807 lines
33 KiB
Plaintext
807 lines
33 KiB
Plaintext
section \<open> SI Units \<close>
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theory Units
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imports Groups_mult
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(* HOL.Fields *)
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HOL.Transcendental
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"HOL-Eisbach.Eisbach"
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begin
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text\<open>
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The International System of Units (SI, abbreviated from the French
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Système international (d'unités)) is the modern form of the metric
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system and is the most widely used system of measurement. It comprises
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a coherent system of units of measurement built on seven base units,
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which are the second, metre, kilogram, ampere, kelvin, mole, candela,
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and a set of twenty prefixes to the unit names and unit symbols that
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may be used when specifying multiples and fractions of the units.
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The system also specifies names for 22 derived units, such as lumen and
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watt, for other common physical quantities.
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(cited from \<^url>\<open>https://en.wikipedia.org/wiki/International_System_of_Units\<close>).\<close>
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text\<open> This is an attempt to model the system and its derived entities (cf.
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\<^url>\<open>https://www.quora.com/What-are-examples-of-SI-units\<close>) in Isabelle/HOL.
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The design objective are twofold (and for the case of Isabelle somewhat
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contradictory, see below)
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The construction proceeds in three phases:
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\<^enum> We construct a generic type \<^theory_text>\<open>SI_domain\<close> which is basically an "inner representation" or
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"semantic domain" of all SI types. Since SI-types have an interpretation in this domain,
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it serves to give semantics to type-constructors by operations on this domain, too.
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We construct a multiplicative group on it.
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\<^enum> From \<^theory_text>\<open>SI_domain\<close> we build a \<^theory_text>\<open>'a SI_tagged_domain\<close> types, i.e. a polymorphic family of values
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tagged with values from \<^theory_text>\<open>SI_domain\<close>. We construct multiplicative and additive
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groups over it.
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\<^enum> We construct a type-class characterizing SI - type expressions
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and types tagged with SI - type expressions; this construction paves the
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way to overloaded interpretation functions from SI type-expressions to
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\<close>
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section\<open>The Domains of SI types and SI-tagged types\<close>
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subsection \<open> The \<^theory_text>\<open>SI_domain\<close>-type and its operations \<close>
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text \<open> An SI unit associates with each of the seven base unit an integer that denotes the power
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to which it is raised. We use a record to represent this 7-tuple, to enable code generation. \<close>
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record SI_domain =
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Seconds :: int
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Meters :: int
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Kilograms :: int
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Amperes :: int
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Kelvins :: int
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Moles :: int
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Candelas :: int
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text \<open> We define a commutative monoid for SI units. \<close>
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instantiation SI_domain_ext :: (one) one
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begin
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\<comment> \<open> Here, $1$ is the dimensionless unit \<close>
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definition one_SI_domain_ext :: "'a SI_domain_ext"
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where [code_unfold]: "1 = \<lparr> Seconds = 0, Meters = 0, Kilograms = 0, Amperes = 0
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, Kelvins = 0, Moles = 0, Candelas = 0, \<dots> = 1 \<rparr>"
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instance ..
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end
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instantiation SI_domain_ext :: (times) times
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begin
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\<comment> \<open> Multiplication is defined by adding together the powers \<close>
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definition times_SI_domain_ext :: "'a SI_domain_ext \<Rightarrow> 'a SI_domain_ext \<Rightarrow> 'a SI_domain_ext"
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where [code_unfold]:
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"x * y = \<lparr> Seconds = Seconds x + Seconds y, Meters = Meters x + Meters y
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, Kilograms = Kilograms x + Kilograms y, Amperes = Amperes x + Amperes y
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, Kelvins = Kelvins x + Kelvins y, Moles = Moles x + Moles y
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, Candelas = Candelas x + Candelas y, \<dots> = more x * more y \<rparr>"
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instance ..
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end
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instance SI_domain_ext :: (comm_monoid_mult) comm_monoid_mult
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proof
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fix a b c :: "'a SI_domain_ext"
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show "a * b * c = a * (b * c)"
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by (simp add: times_SI_domain_ext_def mult.assoc)
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show "a * b = b * a"
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by (simp add: times_SI_domain_ext_def mult.commute)
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show "1 * a = a"
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by (simp add: times_SI_domain_ext_def one_SI_domain_ext_def)
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qed
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text \<open> We also define the inverse and division operations, and an abelian group. \<close>
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instantiation SI_domain_ext :: ("{times,inverse}") inverse
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begin
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definition inverse_SI_domain_ext :: "'a SI_domain_ext \<Rightarrow> 'a SI_domain_ext"
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where [code_unfold]:
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"inverse x = \<lparr> Seconds = - Seconds x , Meters = - Meters x
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, Kilograms = - Kilograms x, Amperes = - Amperes x
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, Kelvins = - Kelvins x, Moles = - Moles x
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, Candelas = - Candelas x, \<dots> = inverse (more x) \<rparr>"
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definition divide_SI_domain_ext :: "'a SI_domain_ext \<Rightarrow> 'a SI_domain_ext \<Rightarrow> 'a SI_domain_ext"
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where [code_unfold]:
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"divide_SI_domain_ext x y = x * (inverse y)"
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instance ..
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end
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print_classes
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instance SI_domain_ext :: (ab_group_mult) ab_group_mult
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proof
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fix a b :: "'a SI_domain_ext"
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show "inverse a \<cdot> a = 1"
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by (simp add: inverse_SI_domain_ext_def times_SI_domain_ext_def one_SI_domain_ext_def)
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show "a \<cdot> inverse b = a div b"
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by (simp add: divide_SI_domain_ext_def)
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qed
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instance SI_domain_ext :: (field) field
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proof
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fix a b :: "'a SI_domain_ext"
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show "inverse a \<cdot> a = 1"
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sledgehammer
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by (simp add: inverse_SI_domain_ext_def times_SI_domain_ext_def one_SI_domain_ext_def)
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show "a \<cdot> inverse b = a div b"
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by (simp add: divide_SI_domain_ext_def)
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qed
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subsection \<open> The \<^theory_text>\<open>SI_tagged_domain\<close>-type and its operations \<close>
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record 'a SI_tagged_domain =
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factor :: 'a
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unit :: SI_domain
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instantiation unit :: comm_monoid_add
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begin
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definition "zero_unit = ()"
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definition "plus_unit (x::unit) (y::unit) = ()"
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instance proof qed (simp_all)
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end
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instantiation unit :: comm_monoid_mult
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begin
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definition "one_unit = ()"
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definition "times_unit (x::unit) (y::unit) = ()"
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instance proof qed (simp_all)
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end
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instantiation unit :: inverse
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begin
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definition "inverse_unit (x::unit) = ()"
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definition "divide_unit (x::unit) (y::unit) = ()"
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instance ..
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end
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instantiation SI_tagged_domain_ext :: (times, times) times
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begin
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definition "times_SI_tagged_domain_ext x y = \<lparr> factor = factor x \<cdot> factor y, unit = unit x \<cdot> unit y, \<dots> = more x \<cdot> more y \<rparr>"
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instance ..
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end
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instantiation SI_tagged_domain_ext :: (zero, zero) zero
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begin
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definition "zero_SI_tagged_domain_ext = \<lparr> factor = 0, unit = 1, \<dots> = 0 \<rparr>"
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instance ..
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end
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instantiation SI_tagged_domain_ext :: (one, one) one
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begin
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definition "one_SI_tagged_domain_ext = \<lparr> factor = 1, unit = 1, \<dots> = 1 \<rparr>"
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instance ..
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end
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instantiation SI_tagged_domain_ext :: (inverse, inverse) inverse
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begin
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definition "inverse_SI_tagged_domain_ext x = \<lparr> factor = inverse (factor x), unit = inverse (unit x), \<dots> = inverse (more x) \<rparr>"
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definition "divide_SI_tagged_domain_ext x y = \<lparr> factor = factor x / factor y, unit = unit x / unit y, \<dots> = more x / more y \<rparr>"
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instance ..
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end
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instance SI_tagged_domain_ext :: (comm_monoid_mult, comm_monoid_mult) comm_monoid_mult
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by (intro_classes, simp_all add: one_SI_tagged_domain_ext_def
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times_SI_tagged_domain_ext_def mult.assoc, simp add: mult.commute)
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text \<open> A base unit is an SI_tagged_domain unit here precisely one unit has power 1. \<close>
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definition is_BaseUnit :: "SI_domain \<Rightarrow> bool" where
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"is_BaseUnit u = (\<exists> n. u = 1\<lparr>Meters := n\<rparr> \<or> u = 1\<lparr>Kilograms := n\<rparr> \<or> u = 1\<lparr>Seconds := n\<rparr>
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\<or> u = 1\<lparr>Amperes := n\<rparr> \<or> u = 1\<lparr>Kelvins := n\<rparr> \<or> u = 1\<lparr>Moles := n\<rparr>
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\<or> u = 1\<lparr>Candelas := n\<rparr>)"
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section\<open>The Syntax and Semantics of SI types and SI-tagged types\<close>
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subsection \<open> Basic SI-types \<close>
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text \<open> We provide a syntax for type-expressions; The definition of
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the basic type constructors is straight-forward via a one-elementary set. \<close>
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typedef meter = "UNIV :: unit set" .. setup_lifting type_definition_meter
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typedef kilogram = "UNIV :: unit set" .. setup_lifting type_definition_kilogram
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typedef second = "UNIV :: unit set" .. setup_lifting type_definition_second
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typedef ampere = "UNIV :: unit set" .. setup_lifting type_definition_ampere
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typedef kelvin = "UNIV :: unit set" .. setup_lifting type_definition_kelvin
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typedef mole = "UNIV :: unit set" .. setup_lifting type_definition_mole
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typedef candela = "UNIV :: unit set" .. setup_lifting type_definition_candela
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subsection \<open> SI-type expressions and SI-type interpretation \<close>
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text \<open> The case for the construction of the multiplicative and inverse operators requires ---
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thus, the unary and binary operators on our SI type language --- require that their arguments
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are restricted to the set of SI-type expressions.
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The mechanism in Isabelle to characterize a certain sub-class of Isabelle-type expressions
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are \<^emph>\<open>type classes\<close>. We therefore need such a sub-class; for reasons of convenience,
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we combine its construction also with the "semantics" of SI types in terms of
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@{typ SI_domain}. \<close>
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subsubsection \<open> SI-type expression definition as type-class \<close>
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class si_type = finite +
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fixes si_sem :: "'a itself \<Rightarrow> SI_domain"
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assumes unitary_unit_pres: "card (UNIV::'a set) = 1"
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syntax
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"_SI" :: "type \<Rightarrow> logic" ("SI'(_')")
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translations
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"SI('a)" == "CONST si_sem TYPE('a)"
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text \<open> The sub-set of basic SI type expressions can be characterized by the following
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operation: \<close>
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class si_baseunit = si_type +
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assumes is_BaseUnit: "is_BaseUnit SI('a)"
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subsubsection \<open> SI base type constructors \<close>
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text\<open>We embed the basic SI types into the SI type expressions: \<close>
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declare [[show_sorts]]
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instantiation meter :: si_baseunit
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begin
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definition si_sem_meter :: "meter itself \<Rightarrow> SI_domain" where "si_sem_meter x = 1\<lparr>Meters := 1\<rparr>"
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instance by (intro_classes, auto simp add: si_sem_meter_def is_BaseUnit_def, (transfer, simp)+)
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end
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instantiation kilogram :: si_baseunit
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begin
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definition si_sem_kilogram :: "kilogram itself \<Rightarrow> SI_domain" where "si_sem_kilogram x = 1\<lparr>Kilograms := 1\<rparr>"
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instance by (intro_classes, auto simp add: si_sem_kilogram_def is_BaseUnit_def, (transfer, simp)+)
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end
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instantiation second :: si_baseunit
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begin
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definition si_sem_second :: "second itself \<Rightarrow> SI_domain" where "si_sem_second x = 1\<lparr>Seconds := 1\<rparr>"
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instance by (intro_classes, auto simp add: si_sem_second_def is_BaseUnit_def, (transfer, simp)+)
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end
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instantiation ampere :: si_baseunit
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begin
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definition si_sem_ampere :: "ampere itself \<Rightarrow> SI_domain" where "si_sem_ampere x = 1\<lparr>Amperes := 1\<rparr>"
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instance by (intro_classes, auto simp add: si_sem_ampere_def is_BaseUnit_def, (transfer, simp)+)
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end
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instantiation kelvin :: si_baseunit
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begin
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definition si_sem_kelvin :: "kelvin itself \<Rightarrow> SI_domain" where "si_sem_kelvin x = 1\<lparr>Kelvins := 1\<rparr>"
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instance by (intro_classes, auto simp add: si_sem_kelvin_def is_BaseUnit_def, (transfer, simp)+)
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end
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instantiation mole :: si_baseunit
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begin
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definition si_sem_mole :: "mole itself \<Rightarrow> SI_domain" where "si_sem_mole x = 1\<lparr>Moles := 1\<rparr>"
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instance by (intro_classes, auto simp add: si_sem_mole_def is_BaseUnit_def, (transfer, simp)+)
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end
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instantiation candela :: si_baseunit
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begin
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definition si_sem_candela :: "candela itself \<Rightarrow> SI_domain" where "si_sem_candela x = 1\<lparr>Candelas := 1\<rparr>"
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instance by (intro_classes, auto simp add: si_sem_candela_def is_BaseUnit_def, (transfer, simp)+)
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end
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lemma [simp] : "is_BaseUnit SI(meter)" by(simp add: Units.si_baseunit_class.is_BaseUnit)
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lemma [simp] : "is_BaseUnit SI(kilogram)" by(simp add: Units.si_baseunit_class.is_BaseUnit)
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lemma [simp] : "is_BaseUnit SI(second)" by(simp add: Units.si_baseunit_class.is_BaseUnit)
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lemma [simp] : "is_BaseUnit SI(ampere)" by(simp add: Units.si_baseunit_class.is_BaseUnit)
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lemma [simp] : "is_BaseUnit SI(kelvin)" by(simp add: Units.si_baseunit_class.is_BaseUnit)
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lemma [simp] : "is_BaseUnit SI(mole)" by(simp add: Units.si_baseunit_class.is_BaseUnit)
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lemma [simp] : "is_BaseUnit SI(candela)" by(simp add: Units.si_baseunit_class.is_BaseUnit)
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subsubsection \<open> Higher SI Type Constructors: Inner Product and Inverse \<close>
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text\<open>On the class of SI-types (in which we have already inserted the base SI types),
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the definitions of the type constructors for inner product and inverse is straight) forward.\<close>
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typedef ('a::si_type, 'b::si_type) UnitTimes (infixl "\<cdot>" 69) = "UNIV :: unit set" ..
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setup_lifting type_definition_UnitTimes
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text \<open> We can prove that multiplication of two SI types yields an SI type. \<close>
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instantiation UnitTimes :: (si_type, si_type) si_type
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begin
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definition si_sem_UnitTimes :: "('a \<cdot> 'b) itself \<Rightarrow> SI_domain" where
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"si_sem_UnitTimes x = SI('a) * SI('b)"
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instance by (intro_classes, simp_all add: si_sem_UnitTimes_def, (transfer, simp)+)
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end
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text \<open> Similarly, we define division of two SI types and prove that SI types are closed under this. \<close>
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typedef 'a UnitInv ("(_\<^sup>-\<^sup>1)" [999] 999) = "UNIV :: unit set" ..
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setup_lifting type_definition_UnitInv
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instantiation UnitInv :: (si_type) si_type
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begin
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definition si_sem_UnitInv :: "('a\<^sup>-\<^sup>1) itself \<Rightarrow> SI_domain" where
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"si_sem_UnitInv x = inverse SI('a)"
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instance by (intro_classes, simp_all add: si_sem_UnitInv_def, (transfer, simp)+)
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end
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subsubsection \<open> Syntactic Support for SI type expressions. \<close>
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text\<open>A number of type-synonyms allow for more compact notation: \<close>
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type_synonym ('a, 'b) UnitDiv = "'a \<cdot> ('b\<^sup>-\<^sup>1)" (infixl "'/" 69)
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type_synonym 'a UnitSquare = "'a \<cdot> 'a" ("(_)\<^sup>2" [999] 999)
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type_synonym 'a UnitCube = "'a \<cdot> 'a \<cdot> 'a" ("(_)\<^sup>3" [999] 999)
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type_synonym 'a UnitInvSquare = "('a\<^sup>2)\<^sup>-\<^sup>1" ("(_)\<^sup>-\<^sup>2" [999] 999)
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type_synonym 'a UnitInvCube = "('a\<^sup>3)\<^sup>-\<^sup>1" ("(_)\<^sup>-\<^sup>3" [999] 999)
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translations (type) "'a\<^sup>-\<^sup>2" <= (type) "('a\<^sup>2)\<^sup>-\<^sup>1"
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translations (type) "'a\<^sup>-\<^sup>3" <= (type) "('a\<^sup>3)\<^sup>-\<^sup>1"
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print_translation \<open>
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[(@{type_syntax UnitTimes},
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fn ctx => fn [a, b] =>
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if (a = b)
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then Const (@{type_syntax UnitSquare}, dummyT) $ a
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else case a of
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Const (@{type_syntax UnitTimes}, _) $ a1 $ a2 =>
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if (a1 = a2 andalso a2 = b)
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then Const (@{type_syntax UnitCube}, dummyT) $ a1
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else raise Match |
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Const (@{type_syntax UnitSquare}, _) $ a' =>
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if (@{print} a' = b)
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then Const (@{type_syntax UnitCube}, dummyT) $ a'
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else raise Match |
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_ => raise Match)]
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\<close>
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subsection \<open> SI Tagged Types \<close>
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text\<open>We 'lift' SI type expressions to SI tagged type expressions as follows:\<close>
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typedef (overloaded) ('n, 'u::si_type) Unit ("_[_]" [999,0] 999)
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= "{x :: 'n SI_tagged_domain. unit x = SI('u)}"
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morphisms fromUnit toUnit by (rule_tac x="\<lparr> factor = undefined, unit = SI('u) \<rparr>" in exI, simp)
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text \<open> Coerce values when their units are equivalent \<close>
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definition coerceUnit :: "'u\<^sub>2 itself \<Rightarrow> 'a['u\<^sub>1::si_type] \<Rightarrow> 'a['u\<^sub>2::si_type]" where
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"SI('u\<^sub>1) = SI('u\<^sub>2) \<Longrightarrow> coerceUnit t x = (toUnit (fromUnit x))"
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section\<open>Operations SI-tagged types via their Semantic Domains\<close>
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subsection\<open>Predicates on SI-tagged types\<close>
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text \<open> Two SI types are equivalent if they have the same value-level units. \<close>
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definition Unit_equiv :: "'n['a::si_type] \<Rightarrow> 'n['b::si_type] \<Rightarrow> bool" (infix "\<approx>\<^sub>U" 50) where
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"a \<approx>\<^sub>U b \<longleftrightarrow> fromUnit a = fromUnit b"
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text\<open>This gives us an equivalence, but, unfortunately, not a congruence.\<close>
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lemma Unit_equiv_refl [simp]: "a \<approx>\<^sub>U a"
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by (simp add: Unit_equiv_def)
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lemma Unit_equiv_sym: "a \<approx>\<^sub>U b \<Longrightarrow> b \<approx>\<^sub>U a"
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by (simp add: Unit_equiv_def)
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lemma Unit_equiv_trans: "\<lbrakk> a \<approx>\<^sub>U b; b \<approx>\<^sub>U c \<rbrakk> \<Longrightarrow> a \<approx>\<^sub>U c"
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by (simp add: Unit_equiv_def)
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(* the following series of equivalent statements ... *)
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lemma coerceUnit_eq_iff:
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fixes x :: "'a['u\<^sub>1::si_type]"
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assumes "SI('u\<^sub>1) = SI('u\<^sub>2::si_type)"
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shows "(coerceUnit TYPE('u\<^sub>2) x) \<approx>\<^sub>U x"
|
|
by (metis Unit_equiv_def assms coerceUnit_def fromUnit toUnit_inverse)
|
|
|
|
(* or equivalently *)
|
|
|
|
lemma coerceUnit_eq_iff2:
|
|
fixes x :: "'a['u\<^sub>1::si_type]"
|
|
assumes "SI('u\<^sub>1) = SI('u\<^sub>2::si_type)" and "y = (coerceUnit TYPE('u\<^sub>2) x)"
|
|
shows "x \<approx>\<^sub>U y"
|
|
by (metis Unit_equiv_def assms coerceUnit_def fromUnit toUnit_inverse)
|
|
|
|
lemma updown_eq_iff:
|
|
fixes x :: "'a['u\<^sub>1::si_type]" fixes y :: "'a['u\<^sub>2::si_type]"
|
|
assumes "SI('u\<^sub>1) = SI('u\<^sub>2::si_type)" and "y = (toUnit (fromUnit x))"
|
|
shows "x \<approx>\<^sub>U y"
|
|
by (metis Unit_equiv_def assms fromUnit toUnit_inverse)
|
|
|
|
text\<open>This is more general that \<open>y = x \<Longrightarrow> x \<approx>\<^sub>U y\<close>, since x and y may have different type.\<close>
|
|
|
|
find_theorems "(toUnit (fromUnit _))"
|
|
|
|
|
|
lemma eq_ :
|
|
fixes x :: "'a['u\<^sub>1::si_type]" fixes y :: "'a['u\<^sub>2::si_type]"
|
|
assumes "x \<approx>\<^sub>U y"
|
|
shows "SI('u\<^sub>1) = SI('u\<^sub>2::si_type)"
|
|
by (metis (full_types) Unit_equiv_def assms fromUnit mem_Collect_eq)
|
|
|
|
|
|
subsection\<open>Operations on SI-tagged types\<close>
|
|
|
|
setup_lifting type_definition_Unit
|
|
|
|
lift_definition
|
|
Unit_times :: "('n::times)['a::si_type] \<Rightarrow> 'n['b::si_type] \<Rightarrow> 'n['a \<cdot>'b]" (infixl "*\<^sub>U" 69) is "(*)"
|
|
by (simp add: si_sem_UnitTimes_def times_SI_tagged_domain_ext_def)
|
|
|
|
lift_definition
|
|
Unit_inverse :: "('n::inverse)['a::si_type] \<Rightarrow> 'n['a\<^sup>-\<^sup>1]" ("(_\<^sup>-\<^sup>\<one>)" [999] 999) is "inverse"
|
|
by (simp add: inverse_SI_tagged_domain_ext_def si_sem_UnitInv_def)
|
|
|
|
abbreviation
|
|
Unit_divide :: "('n::{times,inverse})['a::si_type] \<Rightarrow> 'n['b::si_type] \<Rightarrow> 'n['a/'b]" (infixl "'/\<^sub>U" 70) where
|
|
"Unit_divide x y \<equiv> x *\<^sub>U y\<^sup>-\<^sup>\<one>"
|
|
|
|
abbreviation Unit_sq ("(_)\<^sup>\<two>" [999] 999) where "u\<^sup>\<two> \<equiv> u *\<^sub>U u"
|
|
abbreviation Unit_cube ("(_)\<^sup>\<three>" [999] 999) where "u\<^sup>\<three> \<equiv> u *\<^sub>U u *\<^sub>U u"
|
|
|
|
abbreviation Unit_neq_sq ("(_)\<^sup>-\<^sup>\<two>" [999] 999) where "u\<^sup>-\<^sup>\<two> \<equiv> (u\<^sup>\<two>)\<^sup>-\<^sup>\<one>"
|
|
abbreviation Unit_neq_cube ("(_)\<^sup>-\<^sup>\<three>" [999] 999) where "u\<^sup>-\<^sup>\<three> \<equiv> (u\<^sup>\<three>)\<^sup>-\<^sup>\<one>"
|
|
|
|
instantiation Unit :: (zero,si_type) zero
|
|
begin
|
|
lift_definition zero_Unit :: "('a, 'b) Unit" is "\<lparr> factor = 0, unit = SI('b) \<rparr>"
|
|
by simp
|
|
instance ..
|
|
end
|
|
|
|
instantiation Unit :: (one,si_type) one
|
|
begin
|
|
lift_definition one_Unit :: "('a, 'b) Unit" is "\<lparr> factor = 1, unit = SI('b) \<rparr>"
|
|
by simp
|
|
instance ..
|
|
end
|
|
|
|
instantiation Unit :: (plus,si_type) plus
|
|
begin
|
|
lift_definition plus_Unit :: "'a['b] \<Rightarrow> 'a['b] \<Rightarrow> 'a['b]"
|
|
is "\<lambda> x y. \<lparr> factor = factor x + factor y, unit = SI('b) \<rparr>"
|
|
by (simp)
|
|
instance ..
|
|
end
|
|
|
|
instance Unit :: (semigroup_add,si_type) semigroup_add
|
|
by (intro_classes, transfer, simp add: add.assoc)
|
|
|
|
instance Unit :: (ab_semigroup_add,si_type) ab_semigroup_add
|
|
by (intro_classes, transfer, simp add: add.commute)
|
|
|
|
instance Unit :: (monoid_add,si_type) monoid_add
|
|
by (intro_classes; (transfer, simp))
|
|
|
|
instance Unit :: (comm_monoid_add,si_type) comm_monoid_add
|
|
by (intro_classes; transfer, simp)
|
|
|
|
instantiation Unit :: (uminus,si_type) uminus
|
|
begin
|
|
lift_definition uminus_Unit :: "'a['b] \<Rightarrow> 'a['b]"
|
|
is "\<lambda> x. \<lparr> factor = - factor x, unit = unit x \<rparr>" by (simp)
|
|
instance ..
|
|
end
|
|
|
|
instantiation Unit :: (minus,si_type) minus
|
|
begin
|
|
lift_definition minus_Unit :: "'a['b] \<Rightarrow> 'a['b] \<Rightarrow> 'a['b]"
|
|
is "\<lambda> x y. \<lparr> factor = factor x - factor y, unit = unit x \<rparr>" by (simp)
|
|
|
|
instance ..
|
|
end
|
|
|
|
instance Unit :: (numeral,si_type) numeral ..
|
|
|
|
instantiation Unit :: (times,si_type) times
|
|
begin
|
|
lift_definition times_Unit :: "'a['b] \<Rightarrow> 'a['b] \<Rightarrow> 'a['b]"
|
|
is "\<lambda> x y. \<lparr> factor = factor x * factor y, unit = SI('b) \<rparr>"
|
|
by simp
|
|
instance ..
|
|
end
|
|
|
|
instance Unit :: (power,si_type) power ..
|
|
|
|
instance Unit :: (semigroup_mult,si_type) semigroup_mult
|
|
by (intro_classes, transfer, simp add: mult.assoc)
|
|
|
|
instance Unit :: (ab_semigroup_mult,si_type) ab_semigroup_mult
|
|
by (intro_classes, (transfer, simp add: mult.commute))
|
|
|
|
instance Unit :: (comm_semiring,si_type) comm_semiring
|
|
by (intro_classes, transfer, simp add: linordered_field_class.sign_simps(18) mult.commute)
|
|
|
|
instance Unit :: (comm_semiring_0,si_type) comm_semiring_0
|
|
by (intro_classes, (transfer, simp)+)
|
|
|
|
instance Unit :: (comm_monoid_mult,si_type) comm_monoid_mult
|
|
by (intro_classes, (transfer, simp add: mult.commute)+)
|
|
|
|
instance Unit :: (comm_semiring_1,si_type) comm_semiring_1
|
|
by (intro_classes; (transfer, simp add: semiring_normalization_rules(1-8,24)))
|
|
|
|
instantiation Unit :: (divide,si_type) divide
|
|
begin
|
|
lift_definition divide_Unit :: "'a['b] \<Rightarrow> 'a['b] \<Rightarrow> 'a['b]"
|
|
is "\<lambda> x y. \<lparr> factor = factor x div factor y, unit = SI('b) \<rparr>" by simp
|
|
instance ..
|
|
end
|
|
|
|
instantiation Unit :: (inverse,si_type) inverse
|
|
begin
|
|
lift_definition inverse_Unit :: "'a['b] \<Rightarrow> 'a['b]"
|
|
is "\<lambda> x. \<lparr> factor = inverse (factor x), unit = SI('b) \<rparr>" by simp
|
|
instance ..
|
|
end
|
|
|
|
instantiation Unit :: (order,si_type) order
|
|
begin
|
|
lift_definition less_eq_Unit :: "'a['b] \<Rightarrow> 'a['b] \<Rightarrow> bool" is "\<lambda> x y. factor x \<le> factor y" .
|
|
lift_definition less_Unit :: "'a['b] \<Rightarrow> 'a['b] \<Rightarrow> bool" is "\<lambda> x y. factor x < factor y" .
|
|
instance by (intro_classes, (transfer, simp add: less_le_not_le)+)
|
|
end
|
|
|
|
lift_definition mk_unit :: "'a \<Rightarrow> 'u itself \<Rightarrow> ('a::one)['u::si_type]"
|
|
is "\<lambda> n u. \<lparr> factor = n, unit = SI('u) \<rparr>" by simp
|
|
|
|
syntax "_mk_unit" :: "logic \<Rightarrow> type \<Rightarrow> logic" ("UNIT'(_, _')")
|
|
translations "UNIT(n, 'a)" == "CONST mk_unit n TYPE('a)"
|
|
|
|
subsection \<open>Polymorphic Operations for Elementary SI Units \<close>
|
|
|
|
named_theorems si_def
|
|
|
|
definition [si_def]: "meter = UNIT(1, meter)"
|
|
definition [si_def]: "second = UNIT(1, second)"
|
|
definition [si_def]: "kilogram = UNIT(1, kilogram)"
|
|
definition [si_def]: "ampere = UNIT(1, ampere)"
|
|
definition [si_def]: "kelvin = UNIT(1, kelvin)"
|
|
definition [si_def]: "mole = UNIT(1, mole)"
|
|
definition [si_def]: "candela = UNIT(1, candela)"
|
|
|
|
subsubsection \<open>The Projection: Stripping the SI-Tags \<close>
|
|
|
|
definition factorUnit :: "'a['u::si_type] \<Rightarrow> 'a" ("\<lbrakk>_\<rbrakk>\<^sub>U") where
|
|
"factorUnit x = factor (fromUnit x)"
|
|
|
|
|
|
subsubsection \<open>More Operations \<close>
|
|
|
|
lemma unit_eq_iff_factor_eq:
|
|
"x = y \<longleftrightarrow> \<lbrakk>x\<rbrakk>\<^sub>U = \<lbrakk>y\<rbrakk>\<^sub>U"
|
|
by (auto simp add: factorUnit_def, transfer, simp)
|
|
|
|
lemma unit_le_iff_factor_le:
|
|
"x \<le> y \<longleftrightarrow> \<lbrakk>x\<rbrakk>\<^sub>U \<le> \<lbrakk>y\<rbrakk>\<^sub>U"
|
|
by (auto simp add: factorUnit_def; (transfer, simp))
|
|
|
|
lemma factorUnit_zero [si_def]: "\<lbrakk>0\<rbrakk>\<^sub>U = 0"
|
|
by (simp add: factorUnit_def, transfer, simp)
|
|
|
|
lemma factorUnit_one [si_def]: "\<lbrakk>1\<rbrakk>\<^sub>U = 1"
|
|
by (simp add: factorUnit_def, transfer, simp)
|
|
|
|
lemma factorUnit_plus [si_def]: "\<lbrakk>x + y\<rbrakk>\<^sub>U = \<lbrakk>x\<rbrakk>\<^sub>U + \<lbrakk>y\<rbrakk>\<^sub>U"
|
|
by (simp add: factorUnit_def, transfer, simp)
|
|
|
|
lemma factorUnit_times [si_def]: "\<lbrakk>x * y\<rbrakk>\<^sub>U = \<lbrakk>x\<rbrakk>\<^sub>U * \<lbrakk>y\<rbrakk>\<^sub>U"
|
|
by (simp add: factorUnit_def, transfer, simp)
|
|
|
|
lemma factorUnit_div [si_def]: "\<lbrakk>x / y\<rbrakk>\<^sub>U = \<lbrakk>x\<rbrakk>\<^sub>U / \<lbrakk>y\<rbrakk>\<^sub>U"
|
|
by (simp add: factorUnit_def, transfer, simp)
|
|
|
|
lemma factorUnit_numeral [si_def]: "\<lbrakk>numeral n\<rbrakk>\<^sub>U = numeral n"
|
|
apply (induct n, simp_all add: si_def)
|
|
apply (metis factorUnit_plus numeral_code(2))
|
|
apply (metis factorUnit_one factorUnit_plus numeral_code(3))
|
|
done
|
|
|
|
lemma factorUnit_mk [si_def]: "\<lbrakk>UNIT(n, 'u::si_type)\<rbrakk>\<^sub>U = n"
|
|
by (simp add: factorUnit_def, transfer, simp)
|
|
|
|
method si_calc =
|
|
(simp add: unit_eq_iff_factor_eq unit_le_iff_factor_le si_def)
|
|
|
|
section \<open> Some Derived Units \<close>
|
|
|
|
definition "radian = 1 \<cdot> (meter *\<^sub>U meter\<^sup>-\<^sup>\<one>)"
|
|
|
|
definition degree :: "real[meter / meter]" where
|
|
[si_def]: "degree = (2\<cdot>(UNIT(pi,_)) / 180)\<cdot>radian"
|
|
|
|
abbreviation degrees ("_\<degree>" [999] 999) where "n\<degree> \<equiv> n\<cdot>degree"
|
|
|
|
definition [si_def]: "litre = 1/1000 \<cdot> meter\<^sup>\<three>"
|
|
|
|
definition [si_def]: "pint = 0.56826125 \<cdot> litre"
|
|
|
|
definition [si_def]: "milli = UNIT(0.001, _)"
|
|
|
|
definition [si_def]: "kilo = UNIT(1000, _)"
|
|
|
|
definition [si_def]: "hour = 3600 \<cdot> second"
|
|
|
|
abbreviation "tonne \<equiv> kilo\<cdot>kilogram"
|
|
|
|
abbreviation "newton \<equiv> (kilogram *\<^sub>U meter) /\<^sub>U second\<^sup>\<two>"
|
|
|
|
abbreviation "volt \<equiv> kilogram *\<^sub>U meter\<^sup>\<two> *\<^sub>U second\<^sup>-\<^sup>\<three> *\<^sub>U ampere\<^sup>-\<^sup>\<one>"
|
|
|
|
abbreviation "watt \<equiv> kilogram *\<^sub>U meter\<^sup>\<two> *\<^sub>U second\<^sup>-\<^sup>\<three>"
|
|
|
|
abbreviation "joule \<equiv> kilogram *\<^sub>U meter\<^sup>\<two> *\<^sub>U second\<^sup>-\<^sup>\<two>"
|
|
|
|
text\<open>The full beauty of the approach is perhaps revealed here, with the
|
|
type of a classical three-dimensional gravitation field:\<close>
|
|
type_synonym gravitation_field = "(real\<^sup>3 \<Rightarrow> real\<^sup>3)[meter \<cdot> (second)\<^sup>-\<^sup>2]"
|
|
|
|
section \<open> Tactic Support for SI type expressions. \<close>
|
|
|
|
lemmas [si_def] = Units.si_sem_meter_def Units.si_sem_kilogram_def Units.si_sem_second_def
|
|
Units.si_sem_ampere_def Units.si_sem_kelvin_def Units.si_sem_mole_def
|
|
Units.si_sem_candela_def
|
|
|
|
si_sem_UnitTimes_def si_sem_UnitInv_def
|
|
times_SI_domain_ext_def one_SI_domain_ext_def
|
|
|
|
(* renaming and putting defs into the rewrite set (which is usually not a good idea) *)
|
|
lemma "SI(meter) = 1\<lparr>Meters := 1\<rparr>" by(simp add: si_def)
|
|
lemma "SI(kilogram)= 1\<lparr>Kilograms := 1\<rparr>" by(simp add: si_def)
|
|
lemma "SI(second) = 1\<lparr>Seconds := 1\<rparr> " by(simp add: si_def)
|
|
lemma "SI(ampere) = 1\<lparr>Amperes := 1\<rparr>" by(simp add: si_def)
|
|
lemma "SI(kelvin) = 1\<lparr>Kelvins := 1\<rparr> " by(simp add: si_def)
|
|
lemma "SI(mole) = 1\<lparr>Moles := 1\<rparr>" by(simp add: si_def)
|
|
lemma "SI(candela) = 1\<lparr>Candelas := 1\<rparr>" by(simp add: si_def)
|
|
|
|
lemma "SI(mole \<cdot> kelvin \<cdot> mole) = SI(kelvin \<cdot> (mole)\<^sup>2)" by(simp add: si_def)
|
|
|
|
lemma [si_def]:"fromUnit UNIT(x::'a::one, second) =
|
|
\<lparr>factor = x,
|
|
unit = \<lparr>Seconds = 1, Meters = 0, Kilograms = 0, Amperes = 0,
|
|
Kelvins = 0, Moles = 0, Candelas = 0\<rparr>\<rparr>"
|
|
by (simp add: mk_unit.rep_eq one_SI_domain_ext_def si_sem_second_def)
|
|
|
|
lemma [si_def]:"fromUnit UNIT(x::'a::one, meter) =
|
|
\<lparr>factor = x,
|
|
unit = \<lparr>Seconds = 0, Meters = 1, Kilograms = 0, Amperes = 0,
|
|
Kelvins = 0, Moles = 0, Candelas = 0\<rparr>\<rparr>"
|
|
by (simp add: mk_unit.rep_eq one_SI_domain_ext_def si_sem_meter_def)
|
|
|
|
lemma [si_def]:"fromUnit UNIT(x::'a::one, kilogram) =
|
|
\<lparr>factor = x,
|
|
unit = \<lparr>Seconds = 0, Meters = 0, Kilograms = 1, Amperes = 0,
|
|
Kelvins = 0, Moles = 0, Candelas = 0\<rparr>\<rparr>"
|
|
by (simp add: mk_unit.rep_eq one_SI_domain_ext_def si_sem_kilogram_def)
|
|
|
|
lemma [si_def]:"fromUnit UNIT(x::'a::one, ampere) =
|
|
\<lparr>factor = x,
|
|
unit = \<lparr>Seconds = 0, Meters = 0, Kilograms = 0, Amperes = 1,
|
|
Kelvins = 0, Moles = 0, Candelas = 0\<rparr>\<rparr>"
|
|
by (simp add: mk_unit.rep_eq one_SI_domain_ext_def si_sem_ampere_def)
|
|
|
|
lemma [si_def]:"fromUnit UNIT(x::'a::one, kelvin) =
|
|
\<lparr>factor = x,
|
|
unit = \<lparr>Seconds = 0, Meters = 0, Kilograms = 0, Amperes = 0,
|
|
Kelvins = 1, Moles = 0, Candelas = 0\<rparr>\<rparr>"
|
|
by (simp add: mk_unit.rep_eq one_SI_domain_ext_def si_sem_kelvin_def)
|
|
|
|
lemma [si_def]:"fromUnit UNIT(x::'a::one, mole) =
|
|
\<lparr>factor = x,
|
|
unit = \<lparr>Seconds = 0, Meters = 0, Kilograms = 0, Amperes = 0,
|
|
Kelvins = 0, Moles = 1, Candelas = 0\<rparr>\<rparr>"
|
|
by (simp add: mk_unit.rep_eq one_SI_domain_ext_def si_sem_mole_def)
|
|
|
|
lemma [si_def]:"fromUnit UNIT(x::'a::one, candela) =
|
|
\<lparr>factor = x,
|
|
unit = \<lparr>Seconds = 0, Meters = 0, Kilograms = 0, Amperes = 0,
|
|
Kelvins = 0, Moles = 0, Candelas = 1\<rparr>\<rparr>"
|
|
by (simp add: mk_unit.rep_eq one_SI_domain_ext_def si_sem_candela_def)
|
|
|
|
|
|
|
|
|
|
lemma Unit_times_commute:
|
|
fixes X::"'a::{one,ab_semigroup_mult}['b::si_type]"and Y::"'a['c::si_type]"
|
|
shows "X *\<^sub>U Y \<approx>\<^sub>U Y *\<^sub>U X"
|
|
unfolding Unit_equiv_def
|
|
by (simp add: Unit_times.rep_eq linordered_field_class.sign_simps(5) times_SI_tagged_domain_ext_def)
|
|
|
|
text\<open>Observe that this commutation law also commutes the types.\<close>
|
|
|
|
(* just a check that instantiation works for special cases ... *)
|
|
lemma " (UNIT(x, candela) *\<^sub>U UNIT(y::'a::{one,ab_semigroup_mult}, mole))
|
|
\<approx>\<^sub>U (UNIT(y, mole) *\<^sub>U UNIT(x, candela)) "
|
|
by(simp add: Unit_times_commute)
|
|
|
|
|
|
lemma Unit_times_assoc:
|
|
fixes X::"'a::{one,ab_semigroup_mult}['b::si_type]"
|
|
and Y::"'a['c::si_type]"
|
|
and Z::"'a['d::si_type]"
|
|
shows "(X *\<^sub>U Y) *\<^sub>U Z \<approx>\<^sub>U X *\<^sub>U (Y *\<^sub>U Z)"
|
|
unfolding Unit_equiv_def
|
|
by (simp add: Unit_times.rep_eq mult.assoc times_SI_tagged_domain_ext_def)
|
|
|
|
text\<open>The following weak congruences will allow for replacing equivalences in contexts
|
|
built from product and inverse. \<close>
|
|
lemma Unit_times_weak_cong_left:
|
|
fixes X::"'a::{one,ab_semigroup_mult}['b::si_type]"
|
|
and Y::"'a['c::si_type]"
|
|
and Z::"'a['d::si_type]"
|
|
assumes "X \<approx>\<^sub>U Y"
|
|
shows "(X *\<^sub>U Z) \<approx>\<^sub>U (Y *\<^sub>U Z)"
|
|
by (metis Unit_equiv_def Unit_times.rep_eq assms)
|
|
|
|
lemma Unit_times_weak_cong_right:
|
|
fixes X::"'a::{one,ab_semigroup_mult}['b::si_type]"
|
|
and Y::"'a['c::si_type]"
|
|
and Z::"'a['d::si_type]"
|
|
assumes "X \<approx>\<^sub>U Y"
|
|
shows "(Z *\<^sub>U X) \<approx>\<^sub>U (Z *\<^sub>U Y)"
|
|
by (metis Unit_equiv_def Unit_times.rep_eq assms)
|
|
|
|
lemma Unit_inverse_weak_cong:
|
|
fixes X::"'a::{field}['b::si_type]"
|
|
and Y::"'a['c::si_type]"
|
|
assumes "X \<approx>\<^sub>U Y"
|
|
shows "(X)\<^sup>-\<^sup>\<one> \<approx>\<^sub>U (Y)\<^sup>-\<^sup>\<one>"
|
|
unfolding Unit_equiv_def
|
|
by (metis Unit_equiv_def Unit_inverse.rep_eq assms)
|
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text\<open>In order to compute a normal form, we would additionally need these three:\<close>
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(* field ? *)
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lemma Unit_inverse_distrib:
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fixes X::"'a::{field}['b::si_type]"
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and Y::"'a['c::si_type]"
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shows "(X *\<^sub>U Y)\<^sup>-\<^sup>\<one> \<approx>\<^sub>U X\<^sup>-\<^sup>\<one> *\<^sub>U Y\<^sup>-\<^sup>\<one>"
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sorry
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(* field ? *)
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lemma Unit_inverse_inverse:
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fixes X::"'a::{field}['b::si_type]"
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shows "((X)\<^sup>-\<^sup>\<one>)\<^sup>-\<^sup>\<one> \<approx>\<^sub>U X"
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unfolding Unit_equiv_def
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sorry
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(* field ? *)
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lemma Unit_mult_cancel:
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fixes X::"'a::{field}['b::si_type]"
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shows "X /\<^sub>U X \<approx>\<^sub>U 1"
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unfolding Unit_equiv_def
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sorry
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lemma Unit_mult_mult_Left_cancel:
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fixes X::"'a::{field}['b::si_type]"
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shows "(1::'a['b/'b]) *\<^sub>U X \<approx>\<^sub>U X"
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unfolding Unit_equiv_def
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apply transfer
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unfolding Unit_equiv_def
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sorry
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lemma "watt *\<^sub>U hour \<approx>\<^sub>U 3600 *\<^sub>U joule"
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unfolding Unit_equiv_def hour_def
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apply(simp add: Units.Unit_times.rep_eq si_def
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zero_SI_tagged_domain_ext_def times_SI_tagged_domain_ext_def
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inverse_SI_tagged_domain_ext_def
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Unit_inverse_def Unit_times_def)
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find_theorems fromUnit
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oops
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thm Units.Unit.toUnit_inverse
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lemma "watt *\<^sub>U hour \<approx>\<^sub>U 3.6 *\<^sub>U kilo *\<^sub>U joule"
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oops
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end
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