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SimonAllingSimonAlling
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Fixed Curry–Howard dashes
The "Curry-Howard isomorphism" (with a hyphen) would be related to a single person named Curry-Howard, whereas the "Curry–Howard" one (with an endash) is related to Curry and Howard. I've used two hyphens ("--") in LaTeX contexts and literal en dashes ("–") elsewhere. I haven't updated any rendered files/snapshots.
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‎2016/Exercises/Exercises-2016-01-27--28.lhs‎

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sentences, we can also find one using the "undefined" introduction and
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elimination rules? The answer, perhaps surprisingly, is *yes*, as
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long as the functions we write are total. This result is known as
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*the Curry-Howard isomorphism*.
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*the Curry–Howard isomorphism*.
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7. Can we extend the Curry-Howard isomorphism to formulas with ¬? In
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7. Can we extend the Curry–Howard isomorphism to formulas with ¬? In
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other words, is there a type that we could use to define Not p, which
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would work together with pairs, ->, and Either to give a full
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translation of sentential logic?

‎2016/Lectures/Lecture04.lhs‎

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Lecture 4: Proofs and Programs in Mathematics
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=============================================
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1. Curry-Howard isomorphism for quantifiers
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1. Curry–Howard isomorphism for quantifiers
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-------------------------------------------
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In the last [exercises](Exercises/Exercises-2016-01-27--28.lhs) we
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The types (c : Term) -> P c and (c : Term, P c) are *dependent types*,
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and cannot be implemented in Haskell. Nevertheless, they are the
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types for which the Curry-Howard isomorphism works, and understanding
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types for which the Curry–Howard isomorphism works, and understanding
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∀ and ∃ as dependently-typed functions and dependent pairs,
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respectively, can be of great help in proofs, as we will see.
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‎2017/E/W2.lhs‎

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sentences, we can also find one using the "undefined" introduction and
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elimination rules? The answer, perhaps surprisingly, is *yes*, as
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long as the functions we write are total. This result is known as
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*the Curry-Howard isomorphism*.
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*the Curry–Howard isomorphism*.
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7. Can we extend the Curry-Howard isomorphism to formulas with ¬? In
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7. Can we extend the Curry–Howard isomorphism to formulas with ¬? In
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other words, is there a type that we could use to define Not p, which
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would work together with pairs, ->, and Either to give a full
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translation of sentential logic?

‎2018/README.md‎

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* Lectures (Tue. 13-15 and Thu 13-15 in EB. [[TimeEdit]](https://se.timeedit.net/web/chalmers/db1/public/ri157375X39Z06Q5Z46g0Y30y5096Y35Q01gQY5Q55767.html))
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* Introduction: Haskell, complex numbers, syntax, semantics, evaluation, approximation
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* Basic concepts of analysis: sequences, limits, convergence, ...
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* Types and mathematics: logic, quantifiers, proofs and programs, Curry-Howard, ...
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* Types and mathematics: logic, quantifiers, proofs and programs, Curry–Howard, ...
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* Type classes, derivatives, differentiation, calculational proofs
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* Domain Specific Languages and algebraic structures, algebras, homomorphisms
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* Polynomials, series, power series

‎L/02/E2.lhs‎

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The answer, perhaps surprisingly, is \emph{yes}, as long as the
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functions we write are total.
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%
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This result is known as \emph{the Curry-Howard isomorphism}.
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This result is known as \emph{the Curry--Howard isomorphism}.
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\end{exercise}
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\begin{exercise}
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Can we extend the Curry-Howard isomorphism to formulas with |¬|?
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Can we extend the Curry--Howard isomorphism to formulas with |¬|?
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%
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In other words, is there a type that we could use to define |Not p|,
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which would work together with pairs, |->|, and |Either| to give a

‎L/02/Live_2_2.lhs‎

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* Case study: sqrt 2 is not rational [proof by contradiction]
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* Case study: p^q can be rational even when neither p nor q is
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* Case study: the limit of a function [from maths text to types and logic]
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* More Curry-Howard: |Either a b| is |Or a b|, |(a,b)| is |And a b|
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* More Curry–Howard: |Either a b| is |Or a b|, |(a,b)| is |And a b|
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* the tupling transform as example
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* typed quantification
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* pushing negation through other constructors

‎L/02/W02.lhs‎

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|f| is a proof of |Forall x (P(x))| if |f t| is a proof of |P(t)| for all |t|.
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\end{quote}
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\paragraph{Curry-Howard}
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\paragraph{Curry--Howard}
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If we abbreviate ``is a proof'' as |:| and use the Haskell convention
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for function application we get
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%
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This now very much looks like type rules, and that is not a coincidence.
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The \emph{Curry-Howard correspondence} says that we can think of
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The \emph{Curry--Howard correspondence} says that we can think of
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propositions as types and proofs as ``programs''.
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%
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These typing judgements are not part of FOL, but the correspondence is
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used quite a bit in this course to keep track of proofs.
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We can also interpret the simpler binary connectives using the
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Curry-Howard correspondence.
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Curry--Howard correspondence.
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A proof of |And P Q| is a pair of a proof of |P| and a proof of |Q|.
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‎L/mapping-2016-2018.md‎

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# Topics covered in L4 2016:
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* Curry-Howard isomorphism for quantifiers → Chapter 2
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* Curry–Howard isomorphism for quantifiers → Chapter 2
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- Implementing intro and elim rules for implication and quantifiers in
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haskell
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* Proofs and programs → Chapter 2

‎README.md‎

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* Lectures
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* Introduction: Haskell, complex numbers, syntax, semantics, evaluation, approximation
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* Basic concepts of analysis: sequences, limits, convergence, ...
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* Types and mathematics: logic, quantifiers, proofs and programs, Curry-Howard, ...
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* Types and mathematics: logic, quantifiers, proofs and programs, Curry–Howard, ...
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* Type classes, derivatives, differentiation, calculational proofs
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* Domain Specific Languages and algebraic structures, algebras, homomorphisms
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* Polynomials, series, power series

‎admin/Course-memo/DSLsofMath.md‎

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* L = Lectures (Tue 13-15 and Thu 13-15 in EB)
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* Introduction: Haskell, complex numbers, syntax, semantics, evaluation, approximation
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* Basic concepts of analysis: sequences, limits, convergence, ...
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* Types and mathematics: logic, quantifiers, proofs and programs, Curry-Howard, ...
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* Types and mathematics: logic, quantifiers, proofs and programs, Curry–Howard, ...
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* Type classes, derivatives, differentiation, calculational proofs
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* Domain Specific Languages and algebraic structures, algebras, homomorphisms
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* Polynomials, series, power series

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