Introduction to programming with dependent types in Scala
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This document introduces programming with dependent types in Scala. It discusses values and types, functions that relate values to types or types to other types, dependent pairs and fibers, using types for calculations at compile-time, and theorem proving using dependent types.
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![Value functions, type families and dependent types
Value-level function (value depending on another value)
(x: Int) => x * x, (x: String) => x + "ab", . . .
Type family (type depending on another type)
Seq[Int], Seq[Boolean], Seq[String], . . .
List[Int], List[Boolean], List[String], . . .
Dependent type (type depending on value of another type)
Sized[Seq[String], 0], Sized[Seq[String], 1], Sized[Seq[String], 2], . . .
val x: Seq[String] = Seq("a", "b", "c")
val x1: Seq[String] = Seq("a", "b", "c", "d")
import shapeless.Sized // Vector type
val x2 : Sized[Seq[String], 3] = Sized("a", "b", "c")
val x3 : Sized[Seq[String], 3] = Sized("a", "b", "c", "d")
Dmytro Mitin Introduction to programming with dependent types in Scala](https://image.slidesharecdn.com/intro-170404145715/85/Introduction-to-programming-with-dependent-types-in-Scala-3-320.jpg)
![Homotopy type theory (HoTT)
https://homotopytypetheory.org/book/
Fiber bundle
Dependent pair type
mkPair(n, vn) !: Sgma(n !: Nat, Vec(n))
( 3, Sized("a", "b", "c")) : Σ(n : Nat, Sized[Seq[String], n])
Dmytro Mitin Introduction to programming with dependent types in Scala](https://image.slidesharecdn.com/intro-170404145715/85/Introduction-to-programming-with-dependent-types-in-Scala-4-320.jpg)
![Type-level programming
Types
True, False, 0, 1, 2, . . . , Int, Boolean, String, . . .
Values
True, False, 0, 1, 2, . . .
Type-level calculations (compile time)
implicitly[(True# && [False])# || [True] =:= True]
implicitly[( 1# + [ 2])# * [ 3] =:= 9]
Value-level calculations (run time)
(true && false) || true == true
(True.&&(False)).||(True) == True
(1 + 2) * 3 == 9
( 1.+( 2)).*( 3) == 9
Dmytro Mitin Introduction to programming with dependent types in Scala](https://image.slidesharecdn.com/intro-170404145715/85/Introduction-to-programming-with-dependent-types-in-Scala-5-320.jpg)
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All the materials around us are made up of elements. These elements can be broadly divided into two major groups:
Metals
Non-Metals
Each group has its own unique physical and chemical properties. Let's understand them one by one.
Physical Properties
1. Appearance
Metals: Shiny (lustrous). Example: gold, silver, copper.
Non-metals: Dull appearance (except iodine, which is shiny).
2. Hardness
Metals: Generally hard. Example: iron.
Non-metals: Usually soft (except diamond, a form of carbon, which is very hard).
3. State
Metals: Mostly solids at room temperature (except mercury, which is a liquid).
Non-metals: Can be solids, liquids, or gases. Example: oxygen (gas), bromine (liquid), sulphur (solid).
4. Malleability
Metals: Can be hammered into thin sheets (malleable).
Non-metals: Not malleable. They break when hammered (brittle).
5. Ductility
Metals: Can be drawn into wires (ductile).
Non-metals: Not ductile.
6. Conductivity
Metals: Good conductors of heat and electricity.
Non-metals: Poor conductors (except graphite, which is a good conductor).
7. Sonorous Nature
Metals: Produce a ringing sound when struck.
Non-metals: Do not produce sound.
Chemical Properties
1. Reaction with Oxygen
Metals react with oxygen to form metal oxides.
These metal oxides are usually basic.
Non-metals react with oxygen to form non-metallic oxides.
These oxides are usually acidic.
2. Reaction with Water
Metals:
Some react vigorously (e.g., sodium).
Some react slowly (e.g., iron).
Some do not react at all (e.g., gold, silver).
Non-metals: Generally do not react with water.
3. Reaction with Acids
Metals react with acids to produce salt and hydrogen gas.
Non-metals: Do not react with acids.
4. Reaction with Bases
Some non-metals react with bases to form salts, but this is rare.
Metals generally do not react with bases directly (except amphoteric metals like aluminum and zinc).
Displacement Reaction
More reactive metals can displace less reactive metals from their salt solutions.
Uses of Metals
Iron: Making machines, tools, and buildings.
Aluminum: Used in aircraft, utensils.
Copper: Electrical wires.
Gold and Silver: Jewelry.
Zinc: Coating iron to prevent rusting (galvanization).
Uses of Non-Metals
Oxygen: Breathing.
Nitrogen: Fertilizers.
Chlorine: Water purification.
Carbon: Fuel (coal), steel-making (coke).
Iodine: Medicines.
Alloys
An alloy is a mixture of metals or a metal with a non-metal.
Alloys have improved properties like strength, resistance to rusting.
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Key concepts covered include:
✅ Definition and Types of Viscosity: Dynamic vs. Kinematic viscosity, cohesion, and adhesion.
⚙️ Methods of Measuring Viscosity:
Rotary Viscometer
Vibrational Viscometer
Falling Object Method
Capillary Viscometer
🌡️ Factors Affecting Viscosity: Temperature, composition, flow rate.
🩺 Clinical Relevance: Impact of blood viscosity in cardiovascular health.
🌊 Fluid Dynamics: Laminar vs. turbulent flow, Reynolds number.
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Chromatography (adsorption, partition, TLC, etc.)
Electrophoresis (protein/DNA separation)
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The YSPH Virtual Medical Operations Center Briefs (VMOC) were created as a service-learning project by faculty and graduate students at the Yale School of Public Health in response to the 2010 Haiti Earthquake. Each year, the VMOC Briefs are produced by students enrolled in Environmental Health Science Course 581 - Public Health Emergencies: Disaster Planning and Response. These briefs compile diverse information sources – including status reports, maps, news articles, and web content– into a single, easily digestible document that can be widely shared and used interactively. Key features of this report include:
- Comprehensive Overview: Provides situation updates, maps, relevant news, and web resources.
- Accessibility: Designed for easy reading, wide distribution, and interactive use.
- Collaboration: The “unlocked" format enables other responders to share, copy, and adapt seamlessly. The students learn by doing, quickly discovering how and where to find critical information and presenting it in an easily understood manner.
CURRENT CASE COUNT: 817 (As of 05/3/2025)
• Texas: 688 (+20)(62% of these cases are in Gaines County).
• New Mexico: 67 (+1 )(92.4% of the cases are from Eddy County)
• Oklahoma: 16 (+1)
• Kansas: 46 (32% of the cases are from Gray County)
HOSPITALIZATIONS: 97 (+2)
• Texas: 89 (+2) - This is 13.02% of all TX cases.
• New Mexico: 7 - This is 10.6% of all NM cases.
• Kansas: 1 - This is 2.7% of all KS cases.
DEATHS: 3
• Texas: 2 – This is 0.31% of all cases
• New Mexico: 1 – This is 1.54% of all cases
US NATIONAL CASE COUNT: 967 (Confirmed and suspected):
INTERNATIONAL SPREAD (As of 4/2/2025)
• Mexico – 865 (+58)
‒Chihuahua, Mexico: 844 (+58) cases, 3 hospitalizations, 1 fatality
• Canada: 1531 (+270) (This reflects Ontario's Outbreak, which began 11/24)
‒Ontario, Canada – 1243 (+223) cases, 84 hospitalizations.
• Europe: 6,814
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Introduction – What heterocyclic compounds are.
Prefix for heteroatom – How to name the different non-carbon atoms in the ring.
Suffix for heterocyclic compounds – How to finish the name depending on the ring size and type.
Nomenclature rules – Simple rules for naming these compounds the right way.
Common rings – Examples of popular heterocyclic compounds used in real life.
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Introduction to programming with dependent types in Scala
- 1. Introduction to programming with dependent types in Scala Dmytro Mitin https://stepik.org/2294 April 2017 Dmytro Mitin Introduction to programming with dependent types in Scala
- 2. Values and types Types Int, Boolean, String, . . . Values 10, true, "abc", . . . 10 : Int true : Boolean "abc" : String . . . Dmytro Mitin Introduction to programming with dependent types in Scala
- 3. Value functions, type families and dependent types Value-level function (value depending on another value) (x: Int) => x * x, (x: String) => x + "ab", . . . Type family (type depending on another type) Seq[Int], Seq[Boolean], Seq[String], . . . List[Int], List[Boolean], List[String], . . . Dependent type (type depending on value of another type) Sized[Seq[String], 0], Sized[Seq[String], 1], Sized[Seq[String], 2], . . . val x: Seq[String] = Seq("a", "b", "c") val x1: Seq[String] = Seq("a", "b", "c", "d") import shapeless.Sized // Vector type val x2 : Sized[Seq[String], 3] = Sized("a", "b", "c") val x3 : Sized[Seq[String], 3] = Sized("a", "b", "c", "d") Dmytro Mitin Introduction to programming with dependent types in Scala
- 4. Homotopy type theory (HoTT) https://homotopytypetheory.org/book/ Fiber bundle Dependent pair type mkPair(n, vn) !: Sgma(n !: Nat, Vec(n)) ( 3, Sized("a", "b", "c")) : Σ(n : Nat, Sized[Seq[String], n]) Dmytro Mitin Introduction to programming with dependent types in Scala
- 5. Type-level programming Types True, False, 0, 1, 2, . . . , Int, Boolean, String, . . . Values True, False, 0, 1, 2, . . . Type-level calculations (compile time) implicitly[(True# && [False])# || [True] =:= True] implicitly[( 1# + [ 2])# * [ 3] =:= 9] Value-level calculations (run time) (true && false) || true == true (True.&&(False)).||(True) == True (1 + 2) * 3 == 9 ( 1.+( 2)).*( 3) == 9 Dmytro Mitin Introduction to programming with dependent types in Scala
- 6. Theorem proving import provingground. import HoTT. import TLImplicits. import shapeless. val indN assoc = NatInd.induc(n :-> (m ∼>: (k ∼>: ( add(add(n)(m))(k) =:= add(n)(add(m)(k)) )))) val hyp = "(n+m)+k=n+(m+k)" :: m ∼>: k ∼>: ( add(add(n)(m))(k) =:= add(n)(add(m)(k)) ) val assoc = indN assoc(m :∼> (k :∼> add(m)(k).refl))( n :∼> (hyp :-> (m :∼> (k :∼> IdentityTyp.extnslty(succ)(add(add(n)(m))(k)) (add(n)(add(m)(k))) (hyp(m)(k)) )))) assoc !: n ∼>: m ∼>: k ∼>: ( add(add(n)(m))(k) =:= add(n)(add(m)(k)) ) Dmytro Mitin Introduction to programming with dependent types in Scala