Added core physical constants of the 2019 SI standard, dimensionless units, and various proof facilities for support.

src/SI/SI_Constants.thy

@@ -1,13 +1,67 @@
 section \<open> Physical Constants \<close>
 
 theory SI_Constants
-  imports SI_Quantities
+  imports SI_Proof
 begin
 
-abbreviation speed_of_light :: "'a::division_ring[L \<cdot> T\<^sup>-\<^sup>1]" where
+abbreviation "hertz \<equiv> second\<^sup>-\<^sup>\<one>"
+
+abbreviation "radian \<equiv> meter \<^bold>\<cdot> meter\<^sup>-\<^sup>\<one>"
+
+abbreviation "steradian \<equiv> meter\<^sup>\<two> \<^bold>\<cdot> meter\<^sup>-\<^sup>\<two>"
+
+abbreviation "joule \<equiv> kilogram \<^bold>\<cdot> meter\<^sup>\<two> \<^bold>\<cdot>  second\<^sup>-\<^sup>\<two>"
+
+abbreviation "watt \<equiv> kilogram \<^bold>\<cdot> meter\<^sup>\<two> \<^bold>\<cdot> second\<^sup>-\<^sup>\<three>"
+
+abbreviation "coulomb \<equiv> ampere \<^bold>\<cdot> second"
+
+abbreviation "lumen \<equiv> candela \<^bold>\<cdot> steradian"
+
+text \<open> The most general types we support must form a field into which the natural numbers can 
+  be injected. \<close>
+
+default_sort field_char_0
+
+abbreviation (input) caesium_frequency:: "'a[T\<^sup>-\<^sup>1]" ("\<Delta>v\<^sub>C\<^sub>s") where
+  "caesium_frequency \<equiv> 9192631770\<cdot>hertz"
+
+abbreviation speed_of_light :: "'a[L \<cdot> T\<^sup>-\<^sup>1]" where
   "speed_of_light \<equiv> 299792458\<cdot>(meter\<^bold>\<cdot>second\<^sup>-\<^sup>\<one>)"
 
-abbreviation gravitational_constant where
+abbreviation Planck :: "'a[M \<cdot> L\<^sup>2 \<cdot> T\<^sup>-\<^sup>2 \<cdot> T]" where
+  "Planck \<equiv> (6.62607015 \<cdot> 1/(10^34))\<cdot>(joule\<^bold>\<cdot>second)"
+
+abbreviation elementary_charge :: "'a[I \<cdot> T]" where
+  "elementary_charge \<equiv> (1.602176634 \<cdot> 1/(10^19))\<cdot>coulomb"
+
+abbreviation Boltzmann :: "'a[M \<cdot> L\<^sup>2 \<cdot> T\<^sup>-\<^sup>2 \<cdot> \<Theta>\<^sup>-\<^sup>1]" where
+  "Boltzmann \<equiv> (1.380649\<cdot>1/(10^23))\<cdot>(joule \<^bold>/ kelvin)"
+
+abbreviation Avogadro :: "'a[N\<^sup>-\<^sup>1]" where
+"Avogadro \<equiv> 6.02214076\<cdot>(10^23)\<cdot>(mole\<^sup>-\<^sup>\<one>)"
+
+abbreviation max_luminous_frequency :: "'a[T\<^sup>-\<^sup>1]" where
+"max_luminous_frequency \<equiv> (540\<cdot>10^12)\<cdot>hertz"
+
+abbreviation luminous_efficacy :: "'a[J \<cdot> (L\<^sup>2 \<cdot> L\<^sup>-\<^sup>2) \<cdot> (M \<cdot> L\<^sup>2 \<cdot> T\<^sup>-\<^sup>3)\<^sup>-\<^sup>1]" where
+"luminous_efficacy \<equiv> 683\<cdot>(lumen\<^bold>/watt)"
+
+abbreviation gravitational_constant :: "'a[L\<^sup>3 \<cdot> M\<^sup>-\<^sup>1 \<cdot> T\<^sup>-\<^sup>2]" where
   "gravitational_constant \<equiv> (6.6743015 \<cdot> 1/(10 ^ 11)) \<cdot> (meter\<^sup>\<three>\<^bold>\<cdot>kilogram\<^sup>-\<^sup>\<one>\<^bold>\<cdot>second\<^sup>-\<^sup>\<two>)"
 
+thm si_def
+
+theorem Quant_eq_iff_same_dim:
+  "x \<approx>\<^sub>Q y \<longleftrightarrow> x = y"
+  by (transfer, simp)
+
+theorem hertz_definition: "1\<cdot>hertz = \<Delta>v\<^sub>C\<^sub>s / 9192631770"
+  by (simp add: unit_eq_iff_magn_eq si_def)
+
+theorem second_definition: "1\<cdot>second \<approx>\<^sub>Q (9192631770::_[\<one>]) \<^bold>/ \<Delta>v\<^sub>C\<^sub>s"
+  by (simp add: unit_equiv_iff, simp add: Quant_equiv_def unit_eq_iff_magn_eq si_def)
+
+default_sort type
+
 end

src/SI/SI_Derived.thy

@@ -2,14 +2,18 @@ section \<open> Derived Units\<close>
 
 theory SI_Derived
   imports SI_Prefix
-begin
+begin                                  
 
-definition "radian = 1 \<cdot> (meter \<^bold>\<cdot> meter\<^sup>-\<^sup>\<one>)"                                  
+definition degree :: "'a::{inverse,real_algebra_1}[L/L]" where
+[si_def]: "degree = (2\<cdot>(UNIT(of_real pi,_)) / 180)\<cdot>radian"
 
-definition degree :: "real[L/L]" where
+abbreviation degrees ("_\<degree>" [999] 999) where "n\<degree> \<equiv> n\<cdot>degree"
-[si_def]: "degree = (2\<cdot>(UNIT(pi,_)) / 180)\<cdot>radian"
 
-abbreviation degrees ("_\<degree>" [999] 999) where "n\<degree> \<equiv> n\<cdot>degree" 
+definition degrees_celcius :: "'a::division_ring[\<Theta>] \<Rightarrow> 'a[\<Theta>]" ("_\<degree>C" [999] 999) 
+  where [si_def]: "degrees_celcius x = (x + 273.151)\<cdot>kelvin"
+
+definition degrees_farenheit :: "'a::division_ring[\<Theta>] \<Rightarrow> 'a[\<Theta>]" ("_\<degree>F" [999] 999)
+  where [si_def]: "degrees_farenheit x = (x + 459.67)\<cdot>5/9\<cdot>kelvin"
 
 definition [si_def]: "litre = 1/1000 \<cdot> meter\<^sup>\<three>"
 
@@ -17,15 +21,20 @@ definition [si_def]: "pint = 0.56826125 \<cdot> litre"
 
 definition [si_def]: "hour = 3600 \<cdot> second"
 
+definition [si_def]: "gram = milli\<cdot>kilogram"
+
 abbreviation "tonne \<equiv> kilo\<cdot>kilogram"
 
 abbreviation "newton \<equiv> (kilogram \<^bold>\<cdot> meter) \<^bold>/ second\<^sup>\<two>"
 
 abbreviation "volt \<equiv> kilogram \<^bold>\<cdot> meter\<^sup>\<two> \<^bold>\<cdot> second\<^sup>-\<^sup>\<three> \<^bold>\<cdot> ampere\<^sup>-\<^sup>\<one>"
 
-abbreviation "watt \<equiv> kilogram \<^bold>\<cdot> meter\<^sup>\<two> \<^bold>\<cdot> second\<^sup>-\<^sup>\<three>"
+definition "inch = 25.5\<cdot>milli\<cdot>meter"
+
+definition "foot = 0.3048\<cdot>meter"
+
+definition "yard = 0.9144\<cdot>meter"
 
-abbreviation "joule \<equiv> kilogram \<^bold>\<cdot> meter\<^sup>\<two> \<^bold>\<cdot>  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>

src/SI/SI_Prefix.thy

@@ -1,7 +1,7 @@
 section \<open> SI Prefixes \<close>
 
 theory SI_Prefix
-  imports SI_Quantities
+  imports SI_Constants
 begin
 
 definition [si_def]: "giga = UNIT(1000000000, _)"

src/SI/SI_Proof.thy

@@ -1,12 +1,12 @@
 theory SI_Proof
-  imports SI_Derived
+  imports SI_Quantities
 begin
 
 section \<open> Tactic Support for SI type expressions. \<close>
 
 lemmas [si_def] =  si_sem_Length_def si_sem_Mass_def si_sem_Time_def 
                    si_sem_Current_def si_sem_Temperature_def si_sem_Amount_def
-                   si_sem_Intensity_def 
+                   si_sem_Intensity_def si_sem_NoDimension_def
 
                    si_sem_UnitTimes_def si_sem_UnitInv_def
                    times_Unit_ext_def one_Unit_ext_def
@@ -19,48 +19,49 @@ lemma "SI(I)  = 1\<lparr>Amperes := 1\<rparr>"    by(simp add: si_def)
 lemma "SI(\<Theta>)  = 1\<lparr>Kelvins := 1\<rparr> "   by(simp add: si_def)
 lemma "SI(N)  = 1\<lparr>Moles := 1\<rparr>"      by(simp add: si_def)
 lemma "SI(J)  = 1\<lparr>Candelas := 1\<rparr>"   by(simp add: si_def)
+lemma "SI(\<one>)  = 1"                 by(simp add: si_def)
 
 lemma "SI(N \<cdot> \<Theta> \<cdot> N) = SI(\<Theta> \<cdot> N\<^sup>2)" by(simp add: si_def)
 
 lemma [si_def]:"fromUnit UNIT(x::'a::one, Time) = 
                   \<lparr>magn = x,
-                   unit = \<lparr>Seconds = 1, Meters = 0, Kilograms = 0, Amperes = 0, 
+                   unit = \<lparr>Meters = 0, Kilograms = 0, Seconds = 1, Amperes = 0, 
                            Kelvins = 0, Moles = 0, Candelas = 0\<rparr>\<rparr>"
   by (simp add: mk_unit.rep_eq one_Unit_ext_def si_sem_Time_def)
 
 lemma [si_def]:"fromUnit UNIT(x::'a::one, Length) = 
                   \<lparr>magn = x,
-                   unit = \<lparr>Seconds = 0, Meters = 1, Kilograms = 0, Amperes = 0, 
+                   unit = \<lparr>Meters = 1, Kilograms = 0, Seconds = 0, Amperes = 0, 
                            Kelvins = 0, Moles = 0, Candelas = 0\<rparr>\<rparr>"
   by (simp add: mk_unit.rep_eq one_Unit_ext_def si_sem_Length_def)
 
 lemma [si_def]:"fromUnit UNIT(x::'a::one, Mass) = 
                   \<lparr>magn = x,
-                   unit = \<lparr>Seconds = 0, Meters = 0, Kilograms = 1, Amperes = 0, 
+                   unit = \<lparr>Meters = 0, Kilograms = 1, Seconds = 0, Amperes = 0, 
                            Kelvins = 0, Moles = 0, Candelas = 0\<rparr>\<rparr>"
   by (simp add: mk_unit.rep_eq one_Unit_ext_def si_sem_Mass_def)
 
 lemma [si_def]:"fromUnit UNIT(x::'a::one, Current) = 
                   \<lparr>magn = x,
-                   unit = \<lparr>Seconds = 0, Meters = 0, Kilograms = 0, Amperes = 1, 
+                   unit = \<lparr>Meters = 0, Kilograms = 0, Seconds = 0, Amperes = 1, 
                            Kelvins = 0, Moles = 0, Candelas = 0\<rparr>\<rparr>"
   by (simp add: mk_unit.rep_eq one_Unit_ext_def si_sem_Current_def)
 
 lemma [si_def]:"fromUnit UNIT(x::'a::one, Temperature) = 
                   \<lparr>magn = x,
-                   unit = \<lparr>Seconds = 0, Meters = 0, Kilograms = 0, Amperes = 0, 
+                   unit = \<lparr>Meters = 0, Kilograms = 0, Seconds = 0, Amperes = 0, 
                            Kelvins = 1, Moles = 0, Candelas = 0\<rparr>\<rparr>"
   by (simp add: mk_unit.rep_eq one_Unit_ext_def si_sem_Temperature_def)
 
 lemma [si_def]:"fromUnit UNIT(x::'a::one, Amount) = 
                   \<lparr>magn = x,
-                   unit = \<lparr>Seconds = 0, Meters = 0, Kilograms = 0, Amperes = 0, 
+                   unit = \<lparr>Meters = 0, Kilograms = 0, Seconds = 0, Amperes = 0, 
                            Kelvins = 0, Moles = 1, Candelas = 0\<rparr>\<rparr>"
   by (simp add: mk_unit.rep_eq one_Unit_ext_def si_sem_Amount_def)
 
 lemma [si_def]:"fromUnit UNIT(x::'a::one, Intensity) = 
                   \<lparr>magn = x,
-                   unit = \<lparr>Seconds = 0, Meters = 0, Kilograms = 0, Amperes = 0, 
+                   unit = \<lparr>Meters = 0, Kilograms = 0, Seconds = 0, Amperes = 0, 
                            Kelvins = 0, Moles = 0, Candelas = 1\<rparr>\<rparr>"
   by (simp add: mk_unit.rep_eq one_Unit_ext_def si_sem_Intensity_def)
 
@@ -109,6 +110,7 @@ lemma Unit_inverse_weak_cong:
   shows   "(X)\<^sup>-\<^sup>\<one>  \<approx>\<^sub>Q  (Y)\<^sup>-\<^sup>\<one>"
   using assms by (transfer, simp)
 
+(*
 text\<open>In order to compute a normal form, we would additionally need these three:\<close>
 (* field ? *)
 lemma Unit_inverse_distrib:
@@ -154,6 +156,6 @@ lemma "watt \<^bold>\<cdot> hour \<approx>\<^sub>Q 3600 \<^bold>\<cdot> joule"
 
 lemma "watt \<^bold>\<cdot> hour \<approx>\<^sub>Q 3.6 \<^bold>\<cdot> kilo \<^bold>\<cdot> joule"
   oops
-
+*)
 
 end

src/SI/SI_Quantities.thy

@@ -18,7 +18,8 @@ lemma Quantity_eq_intro:
 
 instantiation Quantity_ext :: (times, times) times
 begin
-  definition "times_Quantity_ext x y = \<lparr> magn = magn x \<cdot> magn y, unit = unit x \<cdot> unit y, \<dots> = more x \<cdot> more y \<rparr>"
+  definition [si_def]:
+    "times_Quantity_ext x y = \<lparr> magn = magn x \<cdot> magn y, unit = unit x \<cdot> unit y, \<dots> = more x \<cdot> more y \<rparr>"
 instance ..
 end
 
@@ -38,7 +39,7 @@ lemma more_zero [simp]: "more 0 = 0" by (simp add: zero_Quantity_ext_def)
 
 instantiation Quantity_ext :: (one, one) one
 begin
-  definition "one_Quantity_ext = \<lparr> magn = 1, unit = 1, \<dots> = 1 \<rparr>"
+  definition [si_def]: "one_Quantity_ext = \<lparr> magn = 1, unit = 1, \<dots> = 1 \<rparr>"
 instance ..
 end
 
@@ -48,8 +49,8 @@ lemma more_one [simp]: "more 1 = 1" by (simp add: one_Quantity_ext_def)
 
 instantiation Quantity_ext :: (inverse, inverse) inverse
 begin
-  definition "inverse_Quantity_ext x = \<lparr> magn = inverse (magn x), unit = inverse (unit x), \<dots> = inverse (more x) \<rparr>"
+  definition [si_def]: "inverse_Quantity_ext x = \<lparr> magn = inverse (magn x), unit = inverse (unit x), \<dots> = inverse (more x) \<rparr>"
-  definition "divide_Quantity_ext x y = \<lparr> magn = magn x / magn y, unit = unit x / unit y, \<dots> = more x / more y \<rparr>"
+  definition [si_def]: "divide_Quantity_ext x y = \<lparr> magn = magn x / magn y, unit = unit x / unit y, \<dots> = more x / more y \<rparr>"
 instance ..
 end
 
@@ -300,8 +301,6 @@ 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, Length)"
 definition [si_def]: "second   = UNIT(1, Time)"
 definition [si_def]: "kilogram = UNIT(1, Mass)"
@@ -321,6 +320,16 @@ lemma unit_eq_iff_magn_eq:
   "x = y \<longleftrightarrow> \<lbrakk>x\<rbrakk>\<^sub>Q = \<lbrakk>y\<rbrakk>\<^sub>Q"
   by (auto simp add: magnQuant_def, transfer, simp)
 
+lemma unit_equiv_iff:
+  fixes x :: "'a['u\<^sub>1::si_type]" and y :: "'a['u\<^sub>2::si_type]"
+  shows "x \<approx>\<^sub>Q y \<longleftrightarrow> \<lbrakk>x\<rbrakk>\<^sub>Q = \<lbrakk>y\<rbrakk>\<^sub>Q \<and> SI('u\<^sub>1) = SI('u\<^sub>2)"
+proof -
+  have "\<forall>t ta. (ta::'a['u\<^sub>2]) = t \<or> magn (fromUnit ta) \<noteq> magn (fromUnit t)"
+    by (simp add: magnQuant_def unit_eq_iff_magn_eq)
+  then show ?thesis
+    by (metis (full_types) Quant_equiv.rep_eq coerceQuant_eq_iff2 eq_ magnQuant_def)
+qed
+
 lemma unit_le_iff_magn_le:
   "x \<le> y \<longleftrightarrow> \<lbrakk>x\<rbrakk>\<^sub>Q \<le> \<lbrakk>y\<rbrakk>\<^sub>Q"
   by (auto simp add: magnQuant_def; (transfer, simp))
@@ -340,6 +349,12 @@ lemma magnQuant_times [si_def]: "\<lbrakk>x * y\<rbrakk>\<^sub>Q = \<lbrakk>x\<r
 lemma magnQuant_div [si_def]: "\<lbrakk>x / y\<rbrakk>\<^sub>Q = \<lbrakk>x\<rbrakk>\<^sub>Q / \<lbrakk>y\<rbrakk>\<^sub>Q"
   by (simp add: magnQuant_def, transfer, simp)
 
+lemma magnQuant_qinv [si_def]: "\<lbrakk>x\<^sup>-\<^sup>\<one>\<rbrakk>\<^sub>Q = inverse \<lbrakk>x\<rbrakk>\<^sub>Q"
+  by (simp add: magnQuant_def, transfer, simp)
+
+lemma magnQuant_qdiv [si_def]: "\<lbrakk>(x::('a::field)[_]) \<^bold>/ y\<rbrakk>\<^sub>Q = \<lbrakk>x\<rbrakk>\<^sub>Q / \<lbrakk>y\<rbrakk>\<^sub>Q"
+  by (simp add: magnQuant_def, transfer, simp add: field_class.field_divide_inverse)
+
 lemma magnQuant_numeral [si_def]: "\<lbrakk>numeral n\<rbrakk>\<^sub>Q = numeral n"
   apply (induct n, simp_all add: si_def)
   apply (metis magnQuant_plus numeral_code(2))

src/SI/SI_Units.thy

@@ -6,6 +6,8 @@ theory SI_Units
           "HOL-Eisbach.Eisbach"
 begin
 
+named_theorems si_def
+
 text\<open>
 The International System of Units (SI, abbreviated from the French 
 Système international (d'unités)) is the modern form of the metric 
@@ -46,21 +48,28 @@ text \<open> An SI unit associates with each of the seven base unit an integer t
   to which it is raised. We use a record to represent this 7-tuple, to enable code generation. \<close>
 
 record Unit = 
-  Seconds   :: int 
+  Meters    :: int
-  Meters    :: int 
+  Kilograms :: int
-  Kilograms :: int 
+  Seconds   :: int
   Amperes   :: int 
   Kelvins   :: int 
   Moles     :: int
   Candelas  :: int
 
+text \<open> Units scaled by a power of 10 \<close>
+
+type_synonym SUnit = "int Unit_ext"
+
+abbreviation scale :: "SUnit \<Rightarrow> int" where
+"scale \<equiv> more"
+
 text \<open> We define a commutative monoid for SI units. \<close>
 
 instantiation Unit_ext :: (one) one
 begin
   \<comment> \<open> Here, $1$ is the dimensionless unit \<close>
 definition one_Unit_ext :: "'a Unit_ext" 
-  where  [code_unfold]:  "1 = \<lparr> Seconds = 0, Meters = 0, Kilograms = 0, Amperes = 0
+  where  [code_unfold, si_def]:  "1 = \<lparr> Meters = 0, Kilograms = 0, Seconds = 0, Amperes = 0
                                , Kelvins = 0, Moles = 0, Candelas = 0, \<dots> = 1 \<rparr>"
   instance ..
 end
@@ -69,9 +78,9 @@ instantiation Unit_ext :: (times) times
 begin
   \<comment> \<open> Multiplication is defined by adding together the powers \<close>
 definition times_Unit_ext :: "'a Unit_ext \<Rightarrow> 'a Unit_ext \<Rightarrow> 'a Unit_ext" 
-  where [code_unfold]:
+  where [code_unfold, si_def]:
-  "x * y = \<lparr> Seconds = Seconds x + Seconds y, Meters = Meters x + Meters y
+  "x * y = \<lparr> Meters = Meters x + Meters y, Kilograms = Kilograms x + Kilograms y
-           , Kilograms = Kilograms x + Kilograms y, Amperes = Amperes x + Amperes y
+           , Seconds = Seconds x + Seconds y, Amperes = Amperes x + Amperes y
            , Kelvins = Kelvins x + Kelvins y, Moles = Moles x + Moles y
            , Candelas = Candelas x + Candelas y, \<dots> = more x * more y \<rparr>"
   instance ..
@@ -93,9 +102,9 @@ text \<open> We also define the inverse and division operations, and an abelian
 instantiation Unit_ext :: ("{times,inverse}") inverse
 begin
 definition inverse_Unit_ext :: "'a Unit_ext \<Rightarrow> 'a Unit_ext" 
-  where [code_unfold]:
+  where [code_unfold, si_def]:
-  "inverse x = \<lparr> Seconds = - Seconds x , Meters = - Meters x
+  "inverse x = \<lparr> Meters = - Meters x, Kilograms = - Kilograms x
-               , Kilograms = - Kilograms x, Amperes = - Amperes x
+               , Seconds = - Seconds x, Amperes = - Amperes x
                , Kelvins = - Kelvins x, Moles = - Moles x
                , Candelas = - Candelas x, \<dots> = inverse (more x) \<rparr>"
 
@@ -155,13 +164,32 @@ subsection \<open> Basic SI-types \<close>
 text \<open> We provide a syntax for type-expressions; The definition of
 the basic type constructors is straight-forward via a one-elementary set. \<close>
 
-typedef Length ("L")      = "UNIV :: unit set" .. setup_lifting type_definition_Length
+typedef Length      = "UNIV :: unit set" .. setup_lifting type_definition_Length
-typedef Mass ("M")        = "UNIV :: unit set" .. setup_lifting type_definition_Mass
+typedef Mass        = "UNIV :: unit set" .. setup_lifting type_definition_Mass
-typedef Time ("T")        = "UNIV :: unit set" .. setup_lifting type_definition_Time
+typedef Time        = "UNIV :: unit set" .. setup_lifting type_definition_Time
-typedef Current ("I")     = "UNIV :: unit set" .. setup_lifting type_definition_Current
+typedef Current     = "UNIV :: unit set" .. setup_lifting type_definition_Current
-typedef Temperature ("\<Theta>") = "UNIV :: unit set" .. setup_lifting type_definition_Temperature
+typedef Temperature = "UNIV :: unit set" .. setup_lifting type_definition_Temperature
-typedef Amount ("N")      = "UNIV :: unit set" .. setup_lifting type_definition_Amount
+typedef Amount      = "UNIV :: unit set" .. setup_lifting type_definition_Amount
-typedef Intensity ("J")   = "UNIV :: unit set" .. setup_lifting type_definition_Intensity
+typedef Intensity   = "UNIV :: unit set" .. setup_lifting type_definition_Intensity
+typedef NoDimension = "UNIV :: unit set" .. setup_lifting type_definition_NoDimension
+
+type_synonym M = Mass
+type_synonym L = Length
+type_synonym T = Time
+type_synonym I = Current
+type_synonym \<Theta> = Temperature
+type_synonym N = Amount
+type_synonym J = Intensity
+type_notation NoDimension ("\<one>")
+
+translations
+  (type) "M" <= (type) "Mass"
+  (type) "L" <= (type) "Length"
+  (type) "T" <= (type) "Time"
+  (type) "I" <= (type) "Current"
+  (type) "\<Theta>" <= (type) "Temperature"
+  (type) "N" <= (type) "Amount"
+  (type) "J" <= (type) "Intensity"
 
 subsection \<open> SI-type expressions and SI-type interpretation \<close>
 
@@ -238,6 +266,12 @@ begin
 instance by (intro_classes, auto simp add: si_sem_Intensity_def is_BaseUnit_def, (transfer, simp)+)
 end
 
+instantiation NoDimension :: si_type
+begin
+  definition si_sem_NoDimension :: "NoDimension itself \<Rightarrow> Unit" where "si_sem_NoDimension x = 1"
+  instance by (intro_classes, auto simp add: si_sem_NoDimension_def is_BaseUnit_def, (transfer, simp)+)
+end
+
 lemma base_units [simp]: 
   "is_BaseUnit SI(Length)" "is_BaseUnit SI(Mass)" "is_BaseUnit SI(Time)"
   "is_BaseUnit SI(Current)" "is_BaseUnit SI(Temperature)" "is_BaseUnit SI(Amount)"
@@ -313,5 +347,9 @@ type_synonym Frequency = "T\<^sup>-\<^sup>1"
 type_synonym Energy = "L\<^sup>2\<cdot>M\<cdot>T\<^sup>-\<^sup>2"
 type_synonym Power = "L\<^sup>2\<cdot>M\<cdot>T\<^sup>-\<^sup>3"
 type_synonym Force = "L\<cdot>M\<cdot>T\<^sup>-\<^sup>2"
-  
+type_synonym Pressure = "L\<^sup>-\<^sup>1\<cdot>M\<cdot>T\<^sup>-\<^sup>2"
+type_synonym Charge = "I\<cdot>T"
+type_synonym PotentialDifference = "L\<^sup>2\<cdot>M\<cdot>T\<^sup>-\<^sup>3\<cdot>I\<^sup>-\<^sup>1"
+type_synonym Capacitance = "L\<^sup>-\<^sup>2\<cdot>M\<^sup>-\<^sup>1\<cdot>T\<^sup>4\<cdot>I\<^sup>2"
+
 end