Type Theory
 What Are Types? What is Type Theory?
 Type Checking
 Systems
 Martin Lof
 Subtyping
 Intersection Types
 Recursive Types
 Refinement Types
 Quotient Types
 Dependent Types
 Two Level Type Theory
 Data Types
 Syntax and Semantics
 Judgement
 Inference Rules
 Computation Rules
 Pattern Matching
 Metatheory
 Equality
 Logical Relations
 Realizability
 Topics
 Bidirectional Typing Old
 Misc
What Are Types? What is Type Theory?
There isn’t a good answer.
Type Theory is what a type theorist does. Useless answer but most honest. What is physics vs chemistry? There are things firmly in one subject (colliders, tylenol), but others in grey areas (nuclear, atomic spectra). Depends on what the person considers themselves, what journals they submit to.
Some goes for type theory. What is proof theory? Computer Science? Logic? It’s somewhere in all of there.

“The question “What is a type?” has no useful answers. The correct question is “What is a type theory?” – and the answer is: it is a mathematical theory of a system of constructions. The types classify the constructions supported by the theory.”  Bauer
 What We Talk About When We Talk About Types  Nick Benton
 types are not sets  Morris classic
 On the Relationship Between Static Analysis and Type Theory  Neel Krishnaswami “Anyway, my main thoughts on LiYao’s question are in a blog post I wrote a few years ago. Ravi Mangal wanted to know what the difference between static analysis and type theory were, and my answer was that you distinguish the two based on how sad you feel if you can’t prove a substitution theorem.”
Type Checking
https://discuss.ocaml.org/t/writingtypeinferencealgorithmsinocaml/8191 PL zoo grow your own type system inferno a library for unification in type systems hindley miler in applicative style How OCaml type checker works – or what polymorphism and garbage collection have in common The Simple Essence of Algebraic Subtyping: Principal Type Inference with Subtyping Made Easy
Bidirectional Typing
Bidirectional Type Checking  jetbrains
mckinna mcbride gundry. tye inferencei n context https://adam.gundry.co.uk/pub/typeinferenceincontext/#:~:text=We%20consider%20the%20problems%20of,for%20implementing%20type%20inference%20algorithms.
Complete and easy Complete and Easy Bidirectional Typechecking for HigherRank Polymorphism. Dunfield Krishnaswami https://arxiv.org/pdf/1306.6032.pdf
FreezeML https://www.pure.ed.ac.uk/ws/portalfiles/portal/142279783/FreezeML_EMRICH_DOA21022020_AFV.pdf
Nice simple tutorial http://davidchristiansen.dk/tutorials/bidirectional.pdf
pfenning lecture notes https://www.cs.cmu.edu/~fp/courses/15312f04/handouts/15bidirectional.pdf
Hindley Milner
http://dev.stephendiehl.com/fun/006_hindley_milner.html
HindleyMilner Elaboration in Applicative Style  Pottier https://gitlab.inria.fr/fpottier/inferno
type inference as generation and then solving of constraints
types are simple types + a var constructor type schema are forall wrappers around a type let allows you to generalize a type with vars into a schema.
env   e : t   C 
take in env and e, output t and C. Set of contraints. DCG might be nice for the constraints.
% type(+, +,  , ). % constants type(E, int(X), int, []). type(E, bool(X), bool, []). type(E, var(X), Y, []) : lookup(X,E,Y). type(E,ite(E1,E2,E3), tvar(T), C1 ++C2 ++ C3 ++ C) : type(E, E1,T1,C1), type(E, E2,T2,C2)
Systems
https://en.wikipedia.org/wiki/Lambda_cube is a way to classify different subsystems
Typed Expressions
Kind of it’s nice to avoid all the lambda calculus stuff. Consider compound boolean and nat expressions.
#use "topfind";;
#require "ppx_jane";;
type ty = Bool  Nat
type term =
True  False  If of term * term * term  Zero
 Succ of term  Pred of term  IsZero of term [@@deriving sexp, equal, compare]
let term_of_string s = Sexplib.Sexp.of_string s > term_of_sexp
let ex = term_of_string "(if true true false)"
let () = Sexplib.Sexp.pp_hum Format.std_formatter (sexp_of_term ex)
;re
; very goofy since I have no evaluable forms
(declaredatatype Term
((True) (False) (Zero))
)
(declaredatatype Type
((Bool) (Nat))
)
(definefunrec hastype ((t Term) (ty Type)) Bool
(match t (
(True (= ty Bool))
(False (= ty Bool))
(Zero (= ty Nat))
))
)
(push)
(declareconst ty1 Type)
(assert (hastype True ty1))
(checksat)
(getmodel)
(pop)
(definefun value ((x Term)) Bool
true
)
(definefun eval ((x Term) (y Term)) Bool
false
)
(definefun progress () Bool
(forall ((x Term) (ty Type))
(=> (hastype x ty)
(or (exists ((y Term)) (eval x y))
(value x))
)))
(definefun preservation () Bool
(forall ((x Term) (y Term) (ty Type))
(=> (and (hastype x ty) (eval x y))
(hastype y ty)
)))
(push)
(echo "progress")
(assertnot progress)
(checksat)
(pop)
(push)
(echo "preservation")
(assertnot preservation)
(checksat)
(pop)
Is normalization of typed expressions easy? We don’t need logical relations?
Simply Typed lambda Calculus (STLC)
typechecking stlc does not have the binding issues you might expect of a lambda calc
System T
 Harper Class notes
 Girard Proofs and Types
 https://gregorulm.com/godelssystemtinagda/
 Avigad Feferman Dialectica
“Quantifier free theory of functionals of finite type”
System F
https://en.wikipedia.org/wiki/System_F
polymorphic lambda calculus
Lambda2
Terms can depend on types. Big Lambda /
(al types can dpeend on types)
THE SYSTEM F OF VARIABLE TYPES, FIFTEEN YEARS LATER  Girard System F is developed on as a intuitionistic second order arithmetic like how system T is first order.
Data types are definable. Strongly normalizaing
System F as dependent types. Type variables could be runtime data, just a special class of them. In that case, the polymorphic signature of a function is a dependent type.
Reynolds towards a theory of type structure From the get go, considering the implementation of integers across two machines is brought up. A relational intepretation. The “inner” and “outer” view, one which sees representation and one that doesn’t
Types abstraction and parametric polymorphism
Polymorphism is not set theoretic https://inria.hal.science/inria00076261/document http://h2.jaguarpaw.co.uk/posts/polymorphismisnotsettheoretic/
Uniform representation
Polymorphism is set theoretic constructively  Pitts
from typing import Tuple
def interp(typ):
if typ == bool:
{True, False}
if typ == Tuple[A,B]:
{ (a,b) for a in interp(A) for b in interp(B)}
if typ == Callable[A,B]:
# we can't finitely list our universe. Since it needs to include all iterated arrow. Eh. I don't see why really.
Universe = {bool : {True, False}}
def Universe(S): # check if in universe
def Universe(S): # infinite stream.
yield {True, False}
yield
# Or maybe produce Z3 expressions. Hmmm.
Typ = Sort("Type")
def interp(ctx, ty):
if ty == bool:
ForAll([x,x1], interp(x,x1) == (x == x1))
if ty == int:
if ty == TVar:
if ty == Tuple[a,b]:
if ty == Callable[a,b]:
ForAll([x,x1], interp(f,f1) == ForAll([x,x1], interp(x,x1) == interp(f(x), f1(x1))))
if ty == Generic(X,T):
ForAll([x,x1], )
# I can't enumerate ? forall X,
PTS
Pure Type Systems Sorts, axioms
System U
Lambda Cube
 stlc
 stlc+type operators. Lambdas at the type level. Does any term have type $\lambda x. x > x$? I don’t think so. Could add constants like
List
but again, no term has this type.  system F. $\lambda 2$ Adds Big Lambda at at term and forall at type level. Polymorphic types. $\forall x. x > x$
 $F\omega$ $\lambda \omega$
 $lambda P$
Other interesting named systems don’t fit in the cube though.
“computational strength” LF has same computayional strength as stlc. what does that mean?
F Omega
first class type constructors.
CoC
Calculus of constructions
Equi vs iso recursive types
Strong Normalization for the Calculus of constructions  Casinghino
CiC
Whole different ball game (?) Luo’s thesis  An Extended Calculus of Constructions Proof on normalization of CIC and its consistency
Martin Lof
A distinctin can be drawn
LF
$\lambda P$ $\lambda \Pi$ http://www.cs.cmu.edu/~aldrich/sasylf/ Twelf http://twelf.org/wiki/Main_Page https://www.cs.cmu.edu/~fp/papers/cade99.pdf
Abella Beluga
There is a claim that LF (which is a corner of the lambda cube?) has enough power to do lots of good stuff for progamming language metahtoery bu not so much that your life sucks. A Framework for Defining Logics harper honsell plotkin 93 judgements as types
https://en.wikipedia.org/wiki/Logical_framework logical framework can refer to a general notion of a system you encode your logic into or a specific type theory lambda pi system https://en.wikipedia.org/wiki/Dependent_type#First_order_dependent_type_theory
LFSC smtlib3 is also an LF?
Dedukti
dedukti  see term rewriting
https://arxiv.org/abs/2311.07185 Dedukti: a Logical Framework based on the λΠCalculus Modulo Theory https://deducteam.github.io/ https://github.com/Deducteam/lambdapi is the assistant? VS code extension
echo "
(; https://deducteam.github.io/tutorial.html
Doesn't seem valid
#NAME foo.
;)
(; declaring new types. 'static' ;)
A : Type.
Nat: Type.
Z : Nat.
S : Nat > Nat.
(; def and just stating existence are different
Notice the defnition of plus is open.
;)
def 1 := S Z.
def 2 := S 1.
def plus: Nat > Nat > Nat.
[m] plus Z m > m
[n,m] plus (S n) m > S (plus n m).
(; also fine? superfluous thugh. ;)
[n,m] plus n (S m) > S (plus n m).
Bool : Type.
true : Bool.
false : Bool.
def equal : Nat > Nat > Bool.
[] equal Z Z > true
[n,m] equal (S n) (S m) > equal n m
[] equal (S _) Z > false
[] equal Z (S _) > false.
(; avoiding intermediate periods makes it kind of a block of declarations. ;)
IsTrue : Bool > Type.
tt : IsTrue true.
#CHECK tt : IsTrue (equal Z 1).
#ASSERT tt : IsTrue (equal Z Z).
#EVAL (equal Z Z).
def test1 : IsTrue (equal (plus 1 1) 2) := tt.
(; partial functions are fine? They just get stuck. ;)
def pred : Nat > Nat.
[ n : Nat ] pred (S n) > n.
Listn : Nat > Type.
nil : Listn Z.
cons : n: Nat > A > Listn n > Listn (S n).
(; Hmm. but.. I need to take in n?
def length : n : Nat > Listn n > Nat.
[] length _ nil > Z.
[n] length _ (cons _ _ tl) > length tl
;)
(; smart constructors. Huh. So because there is a normalizaton, we're chill.
WHat makes this a 'constructor'. Nothing really.
;)
Int : Type.
def Diff : Nat > Nat > Int.
[m,n] Diff (S m) (S n) > Diff m n.
def abs : Int > Nat.
[n] abs (Diff Z n) > n.
[n] abs (Diff n Z) > n.
def append: n: Nat > Listn n > m: Nat > Listn m > Listn (plus n m).
[l2] append _ nil _ l2 > l2
[n,l1,m,l2,a]
append _ (cons n a l1) m l2 > cons (plus n m) a (append n l1 m l2).
#INFER nil.
#PRINT \"hello world\".
(; partial equality. Neeeeeat. ;)
def A_eq : A > A > Bool.
[a] A_eq a a > true.
(; Thi _does_ extend the power of equal, because stuck terms are also equal now. like pred Z ;)
[ n : Nat ] equal n n > true.
" > /tmp/append.dk
dk check /tmp/append.dk #v
confluence checking. It tries to? Or maybe lambdapi does? “Correct by cnfluence” dk check confluence=CMD
add new rules, confluence checking demonstrates that the new rules are “consistent” wth the old one?
https://terminationportal.org/wiki/TPDB needs to support tpdb format. http://clinformatik.uibk.ac.at/software/csi/ho/
http://cops.uibk.ac.at/?q=hrs confluence problem database
dk meta
rewrites a dedukti file. Could I write an egraph as a file and then rewrite it?
https://github.com/Deducteam/Dedukti/tree/master/examples examples
https://github.com/Deducteam/Dedukti/blob/master/README.md manual
#CHECK t1 == t2.
#CHECKNOT t1 == t2.
#EVAL[10]
#EVAL[SNF]
commands beautify, check, dep, meta, top a toplevel, prune
kontroli is rust version https://arxiv.org/pdf/2102.08766.pdf Safe, Fast, Concurrent Proof Checking for the lambdaPi Calculus Modulo Rewriting
Subtyping
Supposedly is kind of a shit show.
Intersection Types
Recursive Types
iso recursive vs. equirecursive
Refinement Types
Refinement Types: A tutorial Explicit Refinment Types “Instrinsic semantics niterprets typing relationd rathert than terms
 FStar

Liquid Haskell
 PVS?
: initialization(main,main).
: op(900, xfx, user:(::)).
tt :: unit.
main(_) : tt :: Ty, writeln(Ty).
Quotient Types
Cody has alluded to there being accidental stumbling blocks into inconsistency here
Dependent Types
See also
 Coq
 lean
Sometimes we get hung up on that types in some systems are compile time (static) and values are runtime (dynamic).
Certainly dynamic typed languages put the types at runtime. Some statically typed languages persist the type information somewhere at runtime so you can reflect to it if need be,
It is a little more subtle that you can pull values back into compile time. Really, programs themselves are compile time things that are talking about abstract runtime values. The variable x
is kindof sortof a value (I am using terminology in a highly loose nontechnical colloquial sense here).
Dependent types let you mention abstract runtime variables and concrete values in the types. It’s sort of a natural box in the compileruntime, typevalue grid.
calculus of constructions
 http://math.andrej.com/2012/11/08/howtoimplementdependenttypetheoryi/
 a tutorial implementation of dependtly typed lambda calc http://lambdatheultimate.org/node/2340 discussion Simply Easy
 http://augustss.blogspot.com/2007/10/simplereasierinrecentpapersimply.html
 https://eb.host.cs.standrews.ac.uk/writings/thesis.pdf  Brady thesis
 https://assetpdf.scinapse.io/prod/2145108549/2145108549.pdf  idris design and implementation
 minitt
 Weirich OPLSS 2013
 easy as pie
 elaboration zoo
Two Level Type Theory
Data Types
What makes “datatypes” different from the things above? They aren’t. My organization is all fucked.
Schema for allowable definitions.
Introduction Elimination Computation Rule
Inductive Types
nlab https://leanprover.github.io/theorem_proving_in_lean/inductive_types.html
Curious restrictions on their form. positivity restrictions
Parameters vs indices
https://leanprover.github.io/theorem_proving_in_lean/inductive_types.html#axiomaticdetails
parameters stay fixed / opaque in all definitions. indices can be
Inductive Vec a : nat > Type :=
 VNil : Vec a 0
 VCons : forall n, a > Vec a n > Vec a (S n)
.
n is index. a is argument
Equi vs Iso recursive types  https://www.cs.cornell.edu/courses/cs4110/2012fa/lectures/lecture27.pdf Does equality hold on the nose in recursion (equi) or is there some unpacking to do (iso) Coinductive Types
Induction Recursion
https://ncatlab.org/nlab/show/inductiverecursive+type induction recursion induction induction
Syntax and Semantics
Type theory takes syntax more seriously than I ever realize. It’s like drawing a house. Do you draw a box with a triangle over it, or do you really look.
Judgement
These are something you can write on paper that represents “I know this”. This is where your mind meets the paper in some sense.
Examples:
 Has type $\Gamma \turnstile t : T$
 Is well formed
 Is Type $\Gamma \turnstile
 evaluates to (big step/small step) $(1 + 3) \downarrow 4$ or $(1+3) \rightarrow 4$
 Hoare logic step
{true}x := 3{x > 0}

Sequent. Under the assumption A,B hold, A holds. $A,B  A$  Focused Sequent. Kind of fishy that this has intuitive meaning
 Type synthesizes, Type Checks  In bidirictional typing discipline, a type being synthesizable/inferrable from the term vs a type being checkable given both type and term are distinct judgements.
Inference Rules
These are the things drawn with horizontal lines. They take you from
Examples
Concrete computer encodings of judgements and inference rules
 Coq inductive types
 Prolog programs (Twelf is related I think)
 Ocaml/Haskell/python functions. There are many different modes you might pick. You can choose
 Tactics
Computation Rules
I don’t know what this means really. Whatever it means, probably beta reduction (applying lambda functions) is included.
The central idea I think is that if you’re building a theorem prover, it needs to understand how to do something. The hypothesis is that it is useful and flexible for the theorem prover to understand well behaved operations of lambda calculus. Lambda calculus has it’s origin as a study of pure logic. It hopefully is capable of expressing anything that belongs to our intuitive notion of “algorithm” (again this was what it was invented for).
So we might want to build in knowing how to add numbers or do sines or cosines or something into our theorem prover, but we can usefully encode these concepts into the master concepts of lambda calculus.
I am ultimately not convinced or unconvinced this hypothesis holds and wonder if the mainstream of interactive theorem proving isn’t a little too attached to the idea.
Conversion vs Reduction vs Expansion
Coq may have it’s own idiosyncratic naming for different kinds of reduction rules. https://coq.inria.fr/refman/language/core/conversion.html
(fun x => x ) q ~> q
beta conversion(fun x => x) ~> (fun z => z)
alpha conversion, aka dummy index rewritingmatch () with () => z ~> z
match conversionlet x := 3 in x ~> 3
zeta reduction if
Definition foo := 3.
thenfoo ~> 3
delta reduction/unfolding of definitions f ~> (fun x => f x)
, eta conversion. expanding oux ~> match x with tt => x end
Is this right? This is eta forunit
I think eta conversion is the most confusing and puzzling out of all of them. Sometimes eta may refer to only eta expanding functions, but other times it may be referring to the relationship of a data type with it’s match statement.
In some sense functions are very special datatypes. They are not defined via an Inductive
definition. But they are also can be thought of as being kind of similar to these data types. The “constructor” of a function is a lambda binder, and the “eliminator” is an application.
We can encode data types into functions using church like encodings.
match
can be encoded by positing a special function that has a computation rule like match.
Pattern Matching
There is a choice whether to consider pattern matching primitive or not Eliminators dependent pattern matching
Conor Mcbride thesis
a few constructions on constructors Pattern Matching Without K
Metatheory
Many type systems talk about lambda calculus, so the metatheory is intertwined with the rewrite theory of lambda calc.
Mechanized Metatheory
Decidability of type checking
type checking is the question of whether given all the pieces going into the judgement under line of the inference rule, is it decidable to construct a typing tree that produces such a thing. Even this seemingly simple question can be undecidable in some type systems, because higher up in the typing tree you may ask questions that are very difficult. In the presence of nontermination, you probably are encountering a variant of the halting problem. Undecidability of type checking is typically considered to be an undesriable property, but people are willing to entertain the notion.
Princiapl Types
Type inference supposes I give the context and term, but leave out the type, can I construct both the typing tree and the type. Principle types are asking whether there is a unique most general type that can be inferred.
What are principal typings and what are they good for?
“Uniqueness of types” may hold in some system.
Consistency
Progress
Well typed terms can take a step under the eval relation or are values.
A kind of liveness property? Eh. Really it feels kind of “coinductive” to me.
Preservation, Subject Reduction
under evaluation, terms remain well typed.
This can mean evaluation does not cause types to change. These are slightly different statements that may not be equivalent in all systems
Preservation is very reminiscent of verifying that types are an invariant, or that the typing relation is an invariant set.
Normalization
https://www.plslab.org/en/Normalization
normalization by evaluation
type term = Lam of string * term
 Var of string
 App of term * term
type typ = Arr of typ * typ
 Unit
type value = Arr of value Value.Map.t  UnitV
let eval : term > typ > value
Canonicity
The same thing as normalization? https://cs.stackexchange.com/questions/112893/whatdoescanonicitypropertymeanintypetheory
Computation rules
Completeness
Soundness
Soundness is whatever the type system claims to show about evaluation, it shows. Sometimes this is progress + preservation
Head Normal Form
Value
https://ice1000.org/2019/0407Reduction.html
What are possible definitions of “value”?
 a subset of terms
 a separate entity akin to terms
 semantic objects
Some of the distinction and confusion depends on how much you rely on your host meta system.
Neutral
Canonical
Reducible Expression
Curry vs Church style
Extrinsic vs Instrinsic types
Large Elimination
pattern matching on a value to construct a value of Type https://cstheory.stackexchange.com/questions/40339/whatexactlyislargeelimination
bool_to_type : bool > Type := fun b =>
match b with
 true => Unit
 false => Empty
impredicative polymorphism + exlcuded middle + large elimination can only choose 2
Excluded Middle
Axiom classic : forall P:Prop, P \/ ~ P.
Sorts & Universes
https://coq.inria.fr/refman/language/core/sorts.html
 Prop vs Type vs Set  Prop is impredicative. Prop and Set are deleted during extraction? Which make them interestingly different from a metaprogramming perspective.
Prop
SProp
Set
SProp
is Prop with proof irrelevance
Prop is erased at extraction. Prop doesn’t have large elimination. But it is impredicative and can support excluded middle.
Inductive Bool : Prop := T : Bool  F : Bool.
Definition two_elements_Bool (H : T = F) : False :=
match H in (_ = y0) return (match y0 with
 T => True
 F => False
end) with
 eq_refl => I
end.
(* Error: *)
(* Incorrect elimination of "y0" in the inductive type "Bool": *)
(* the return type has sort "Type" while it should be "Prop". *)
(* Elimination of an inductive object of sort Prop *)
(* is not allowed on a predicate in sort Type *)
(* because proofs can be eliminated only to build proofs. *)
[why does coq have prop(https://cstheory.stackexchange.com/questions/21836/whydoescoqhaveprop)]
Is there a difference between Sorts and Universes? https://github.com/jdolson/hottnotes/blob/pdfs/pdfs/notes_week5.pdf Universes are
Algerbaic universes  If the levels of type are a DAG, they can be ordered. So one doesn’t eagerly pick integer levels for Type. Instead you maintain contraints like max{l1,l2} etc.
Impredicative vs Predicative
This is describing allowing some kind of form of self reference I think.
Impredicative types allow quantification over themselves. Prop
Proof Irrelevance
Propositional extensionality implies proof irrelevance https://coq.inria.fr/library/Coq.Logic.ProofIrrelevance.html Axiom proof_irrelevance : forall (P:Prop) (p1 p2:P), p1 = p2.
https://github.com/coq/ceps/pull/32 RFC: Disentangling erasure and proof irrelevance
It feels like proof irrelevance and erasure are related concepts, but like almost everything in type theory there is probably a time and place and level of sophistication to consider them as distinct
Irrelevance, Polymorphism, and Erasure in Type Theory
Inconsistent Combinations
See counterexamples in type theory
 Berardi’s paradox above
 Markov + ? https://ncatlab.org/nlab/show/Markov%27s+principle https://en.wikipedia.org/wiki/Markov%27s_principle
Extensionality
Extensionality is a notion that applies to more than just function and equality types. https://cstheory.stackexchange.com/questions/17636/%CE%B7conversionvsextensionalityinextensionsoflambdacalculus
forall x, f x = g x > f = g forall A B : Prop, A <> B > A = B forall x, x = match x with () > ()
Consider fun x > x = fun x > tt Is this provable?
Some of these are provable
alpha beta eta iota
Computational Content
Heterogenous vs Homogenous equality
Eta equivalence  The dumb expansion? Observational Equivalence
Reduction vs Equivalence
Equality
Extensional vs Intensional
Extensional equality collapses definitional and propositional equality.
Adding a rule
Gamma  t : Id_Type A B

Gamma  A = B
Notice that this is a transition from a typing judgement to a definitional equality judgement. Very different fellas.
Observational Type Theory
observational type theory type theory has an equality type and refl. Extentional type theoru adds reflection, intentional adds eliminators
pavel summarizing some different choices
Judgemental/Definitional / Propositional
https://coq.inria.fr/refman/language/core/conversion.html#conversionrules https://coq.inria.fr/refman/proofs/writingproofs/equality.html
Another way of talking about judgementally equal is if the proof of Prop equality is just eq_refl
. Why is this? Well, type checkeing more or less must look for an inferrance rule with eq_refl : a = b
below the horizontal line. This rule breaks apart this problem into a definitional. Definitional equality is so in the air we breath, it may not even be noted explicitly. Any nonlinear occurence of pattern variables in the inferrence rule must imply it.
Tests. Example Definition mytest : tt = tt := eq_refl.
See if coq accepts it https://coq.vercel.app/scratchpad.html
tt = tt
unit = unit
1 + 1 = 2
x : nat  x + 0 = x
x : nat  0 + x = x
(fun x => x) = (fun x => x)
(fun x => x) = (fun x => tt)
(match tt with tt => tt end) = tt
x : unit  tt = x
x : unit  match x with () => () end = ()
x : unit  match x with () => () end = x
x : unit  match x with () => () end = match x with _ => () end
f  f = (fun x => f x)
eta(fun x => x) tt = tt
beta
Univalence
Implies extensionality
Example unit' = unit
Isomorphisms
HOTT
The homotopy type theory book
Implementing Patch Theories in Homotopy Type Theory https://www.cs.ru.nl/bachelorstheses/2023/Dick_Blankvoort___1056960___Implementing_Patch_Theories_in_Homotopy_Type_Theory.pdf
https://www.youtube.com/watch?v=Pu16iodESTU&list=PL245PKGUDdcN9El9D7DRefwX4c9feiYq&t=594s&ab_channel=JacobNeumann Youtube videos
Older
2018
https://epit2020cnrs.inria.fr/ spring school Type/Space A type Elekemntn/Point t:A type and point may depend on other points x:A1, x2:A2  B(x1,,x2) type
R^n depends on n. n : N  R^n type [a,b] depends on a,b, and a <= b a:R, b:R, p :(a<= b)  [a,b] Equality judgemental equal A === B equal types s ===_A t equal points
Type theory as formal system Gam  A type t A A === B s ===_A t
rules of inference
Base space A a:A  B(x) fiber at x. ttoal space  dpeendent sum
Basic constructions
dependent sum
https://fplab.wordpress.com/2015/05/09/hottforlazyfunctionalprogrammers/
The most straightforward computational attack on homotopy is usuing a trinagulated mesh that I think is fairly intuitive to computer people
Computerizing something involes:
 how to you even represent the thing you want to talk about
 howw do you actually compute answers to interesting questions. I divide these into differewnt classes 1. Just recognize something, i.e.e proof checking or 2. Calculate or find something, i.e. actual numerical computations or proof finding or what have you. 1. Is often an interesting thing to start with since it is easier even if what you really want is 2.
Paths are lists of adjacent vertices.
Using haskell gadts for example, we can encode all the paths and then use the types to enforce that the path is actually connected.
Another persepctive on this is to encopde the proerties of the mesh in logic, like first order logic. Z3 is a way to make this conrete
From the perspective of systems like Coq or agda, these two pictures become the same thing. There systems have the notion of inductive predicates that boil down to defining a datatype like that in Haskell.
It sucks to work with this stuff though.
A second straightforward attack to model homotopy would be to directly model functions from the interval R > R. Reals are floats.
We might try to directly model these things in functions and use functional programming def homo1():
def composepath(p1,p2):
assert(p1(1) == p2(0) )
def res(t):
if t < 0.5:
return p1(2 * t)
else:
return p2(2*t1)
return res
Skeletonizing this path with an open cover. 1 open cover = [0,1) and (0,1]
Or we might try to take a more symbolic approach, like using sympy or some custom dsl. The difference here is tha6t in many languages, lambdas are basically totally opaque. There are advantages and disadvenmtages to this rule. It does not have to be so. Languages with strong metaprogramming paradigms (and I include lisps, Julia, and coq here) let you poke around in the guts to varying degrees.
Reals as floats may be objectionable to your sensibiliuties. Then perhaps a preferable approach may be to use exact reals, numbers that retain the capacity to be computed to arbitrary precision upon request.
Maybe there is more connection to my path thing than I realezied
Higher inductive types are where it is actually at I guess.
https://www.youtube.com/watch?v=A6hXn6QCu0k emily reihl  inifnity categroeis for undergrads, but really it’s about homotopy type thoery
11/2020 https://github.com/HoTTIntro/Agda I screwed up my emacs agda by having a rotten tuareg in it. I think https://github.com/martinescardo/HoTTUFAgdaLectureNotes
https://pure.itu.dk/portal/files/84649948/icfp19main_p164_p.pdf  ICFP cubical agda, a depdently typed programming language with higher inductive types. Interestingly, from Mertberg’s demo, I feel like the emphasis was not on cubical itself. More interested in standard HoTT they just want it to be better
HoTT  quotients supposedly? Fractions? Integers vs naturals. I guess that’s the zeroth. Or maybe True / False modulo True = False? Anders Morberg  cubical agda https://vimeo.com/459020971
data _===_ {A : Set} (x : A) : A > Set where
refl : x == x
funExt
replcace inductive === with paths
{# OPTIONS  cubical #}
open import Cubival.Foundations.Prelude
Equiv
Isomorphism
Univalence
Cubical.Data.Int
Cubical.Data.prod
 I , npoitns i0, i1
apply0 : (A; Type ) > (p : I > A) > A
apply0 A p = p i0
refl' : {A : Type} (x : A) > x \== x  PathP (\ _ > A) x x
refl' x = \i > x
 x \== x h means PathP (\ _ > A) x y
 path over path. depednent paths
cong' : {A B : Type} (f : A > B) {x y : A} > x == y > f x == f y
cong' f p = \ i > f ( p i )
funext' : {A B : Type} (f g : A > B) {x y : A} > ( p: x : A > f x == g y ) > f == g
funext p i = \x > p x i
transport' : {A B : Type} > A == B > A > B
trasnport' p x = transp (\i > p i) i0 x
 another primitive called hcomp
ua' : {A B : Type} > A ~ B > A == B
ua' e = ua e
isToEquiv' : {A B :type} > Iso A B > A ~ B
isToEquiv' e = isoToEquiv e
isoToPath : Iso A B > A == B
isoToPath e = ua (isoToEquiv e)
data Bool : Type where
false : Bool
true : Bool
not : Bool > Bool
not false = true
not true = false
notPath : Bool === Bool
notPath = isoToPath' (iso not not rem rem)
where
rem : (x : Bool) > not (not x) == x
rem false = refl
rem true = refl
transport notPath true  false
sucPath : Int === Int
sucPath = (isoToPath' (iso sucInt predInt sucPred redSuc)
transport ( sucPath . sucPath) (pos 4)
 multisets
data FMSet (A : Type) : Type where
[] : FMSet A
_::_ : A > FMSet A > FMSet A
comm : ( x y : A) (xs : FMSet A) >
x :: y :: xs == y :: x :: xs
trunc : isSet (FMSet A)  UIP
_++_ : {A : Type} > FMSet A > FMSet A > FMSet A
[] ++ ys = ys
(x :: xs) ++ ys = x :: (xs ++ ys)
(comm x y xs i) ++ ys = comm x y (xs ++ ys) i
 can prove all kinds of things : xs ++ [] == xs ...
Cubical.HITs.FiniteMultiSet
unitr++ : {A : Type} (xs : FMSet) > xs ++ [] == xs
unitr++ [] = refl
unitr++ (x :: xs) =
SIP structure idenity principle
Cubical.Algerba.Monoid.Base  reflection
queues, matrices. Huh. representiation independence transportiung proofs.
https://arxiv.org/abs/1701.04391  de moura selsam. good explanation of equality type John Major equality  conor mcbride Doing raw recursor proofs. Can I do it? I feel like it’s kind of straightforward. Begin able to write out exactly where I want the equality applied via a lambda. Could Axiomatize equality rather than use Inductive definition. The pattern match of a HIT must desguar into recursors?
Types are spaces. values are points dependent types are fibrations identity types are paths
Other weirdness: Observational type theory
Describing a concrete simplex as a gadt. How do you make a correct by construction data structure of a path? A list of vertices. Fine. But is there a path between each of them? Ok, list of edges. Do those edges meet?
9/2018
univalence let’s you turn isomorphisms into equality and get isomorphisms out of equalities.
HIT  Higher inductive types. Instead of always needing the data constructor to end with the type itself A , now it can end with =_A also. You can do this using
recursors. Patrern matching is cool,. but it can be desgared into something else. A function that takes a continuation for each case of the type.
focus on inclusion and projection maps of the basic simplex
Cubical Sets.
of of the basic interval.
Simpilical Sets.
By using the basic simplex with its inclusion maps as sort of a index into your complex, you can grab all nsimplices at once. This is a Functor relation.
$latex
Great intro
https://arxiv.org/pdf/0809.4221
You can use a postulate methodology where you slam out all the pieces of your type as axioms
Dan Licata explaining how to use hiding to implement HIT. You export an interface. This let’s you postulate less
https://homotopytypetheory.org/2011/04/23/runningcirclesaroundinyourproofassistant/
Homotopy group of circle is Z
https://homotopytypetheory.org/2012/06/07/asimplerproofthat%CF%80%E2%82%81s%C2%B9isz/
https://www.cs.cmu.edu/~drl/pubs/ls13pi1s1/ls13pi1s1.pdf
You always need streicher K and rewriting turned on. What do these do?
https://stackoverflow.com/questions/39239363/whatisaxiomk/39256457
Axiom K is Uniqeness of Identity Proofs. Default pattern matching allows this. Modified pattern matching doesn’t. The recursor form doesn’t seem to have this problem
uip : {A : Set} > {a : A} > (p q : a ≡ a) > p ≡ q
uip refl refl = refl
https://github.com/HoTT/HoTTAgda
Kind of hard to navigate. Like the stdlib but harder.
Useful start in core. Check out univalence, check out the types folder. Look at the normal types like Bool, Nat, List. Now check out Circle, Torus,
Interesting lectures using cubicaltt
https://github.com/mortberg/cubicaltt/tree/master/lectures
Logical Relations
 History of programming course https://felleisen.org/matthias/7480s21/lectures.html Mentions Amal thesis as being good
 Intro to logical relations
Tait proved strong normalization using them 1967.
Parametricity is supposedly an example also
Chapter (crary) in Advanced Topics in Types by Pierce People keep referencing Mitchell 1996
Logical relations as a generalization of homomorphism  Stump podcast referencing a paper
Realizability
Cody’s suggested book TOC Extensional vs Intensional models? Chapter 1: The type theory and its variants:
 The CIC with universes
 Recursors vs guarded fixpoints
 Predicative variant
 Untyped conversion
 Some additional conversion rules
 Extensionality principles
 Antiextensionality principles
 Coinduction Chapter 2: Theorems about syntax
 Substitution, weakening
 Wellsortedness
 Progress and preservation
 Uniqueness of types(?)
 Properties of normal forms Chapter 3: Extensional models
 The notsosimple set theoretic interpretation
 Models of sets in types
 The effective topos interpretation
 The groupoid interpretation
 Consistency implications Chapter 4: Intensional models
 Realizability interpretations
 The Lambdaset model
 The normalization proof
 Consistency implications Chapter 5: Inconsistent combinations of rules
 Failures of subject reduction
 Failures of termination alone
 Impredicative inconsistencies
 Guard condition and inductive type issues
 Other universe issues
Topics
A collection of programming languages and type systems.
Gradual Typing
Intersection types
subtyping
ALgebraic subtyping https://lptk.github.io/programming/2020/03/26/demystifyingmlsub.html mlsub dolan thesis
union types
Bidirectional Typing Old
Notes on Bidirectional Typing and the Meaning of Horizontal Lines
type var = string
type typ = Unit  Arr of typ * typ
type expr = Var of var  Lam of var * expr 
App of (expr * expr)  TT  Annot of (expr * typ)
type typ_ctx = (var * typ) list
let rec type_check gamma e t = match e,t with
(*  Var v, t > (List.assoc v gamma) = t *)
 TT , Unit > true
 Lam(x,e), Arr(a1,a2) > type_check ((x,a1) :: gamma) e a2
 _, _ > let t' = type_infer gamma e in
t = t'
and type_infer gamma e = match e with
 Var v > List.assoc v gamma
 Annot(e, t) > let c = type_check gamma e t in
if c then t else failwith "Bad annotation"
 App(f, x) > let Arr (a,b) = type_infer gamma f in
let c = type_check gamma x a in
if c then b else failwith "App typecheck failed"
 _ > failwith "can't infer"
(* #use bidi.ml;;
type_check [] TT Unit;;
type_infer [] (Annot (Lam ("x", App(Var "x", TT)), Arr(Arr(Unit,Unit),Unit)));;
type_infer [] (Annot (Lam ("x", TT), Arr(Unit,Unit)));;
*)
% It works but it is not well moded
type(G, var(X), A) : member( X  A , G ). % member is a little wrong.
type(G, ann( E, T), T) : type(G, ann(E,T), T).
type(G, tt, unit).
type(G, lam(X,E), arr(A1,A2)) : type([X  A1  G], E, A2 ).
type(G, app(E1,E2), B) : type(G, E1, arr(A,B)), type(G, E2, A).
%? type([], lam(x,lam(y,var(y))), T).
%T = arr(_6662, arr(_6680, _6680))
Require Import String.
Inductive type :=
 unit
 arr : type > type > type
.
Inductive expr :=
 var : string > expr
 lam : string > expr > expr
 ap : expr > expr > expr
 tt : expr
 ann : expr > type > expr.
Fixpoint type_eq a b :=
match a,b with
 unit, unit => true
 (arr x y), (arr z w) => andb (type_eq x z) ( type_eq y w)
 _, _ => false
end.
Definition ctx := list (string * type).
Require Import List.
Import ListNotations.
Search app.
Fixpoint lookup k l := match l with
 (t,v) :: ls => if String.eqb k t then v else lookup k ls
 [] => unit (* chris mocked me for this *)
end.
(** adequacy *)
Inductive hastype (gamma : ctx) : expr > type > Type :=
 Var : forall x A, lookup x gamma = A > hastype gamma (var x) A
 UnitI : hastype gamma tt unit
 ArrI : forall e x A1 A2, hastype ((x, A1) :: gamma) e A2 >
hastype gamma (lam x e) (arr A1 A2)
 ArrE : forall A B e2 e1,
hastype gamma e1 (arr A B) >
hastype gamma e2 A >
hastype gamma (ap e1 e2) B
 Anno : forall a b e,
hastype gamma e a >
type_eq a b = true >
hastype gamma (ann e a) b
.
Inductive inferstype (g : ctx) : expr > type > Type :=
 Varinf : forall x A, lookup x g = A > inferstype g (var x) A
 ArrEInf : forall A B e2 e1,
inferstype g e1 (arr A B) >
checkstype g e2 A >
inferstype g (ap e1 e2) B
 AnoInf : forall a e, checkstype g e a >
inferstype g (ann e a) a
with checkstype (g : ctx) : expr > type > Type :=
 UnitIc : checkstype g tt unit
 ArrIc : forall e x A1 A2, checkstype ((x, A1) :: g) e A2 >
checkstype g (lam x e) (arr A1 A2)
 Subc : forall A e B, inferstype g e A > type_eq A B = true > checkstype g e B
.
Goal checkstype [] tt unit. eauto using checkstype. Qed.
Goal hastype [] tt unit. eauto using hastype. Qed.
Goal {t & hastype [] tt t}. eauto using hastype. Qed.
Goal checkstype [] (lam "x" tt) (arr unit unit).
eauto using checkstype. Qed.
Goal hastype [] (lam "x" tt) (arr unit unit).
eauto using hastype. Qed.
Goal hastype [] (ap (lam "x" tt) tt) unit.
eauto using hastype. Qed.
Goal inferstype [] (ap
(lam "x" tt)
tt) unit.
eauto using inferstype, checkstype. Abort.
Goal inferstype [] (ap
(ann (lam "x" tt) (arr unit unit))
tt) unit.
eauto using inferstype, checkstype. Qed.
(* eapply ArrEInf.
 eapply AnoInf.
apply ArrIc. apply UnitIc.
 apply UnitIc.
Qed. *)
(* Decision procedures.
Fixpoint inferstype (g:ctx) (e : expr) : option {t : type  inferstype g e t} :=
Fixpoint checkstype (g:ctx) (e : expr) (t : type) : option (checkstype g e t) :=
sound : checkstype > hastype
complete : hastype > checkstype \/ inferstype
*)
From elpi Require Import elpi.
I picked bidirectional typechecking for reading group this week. A lovely survey paper.
Bidirectional typing is a methodology for taking piles of typing rules and making them more algorithmic. This gets into my strike zone, as an outsider to type theory and logic, the notion of inference rules bends me in knots. I don’t know what they mean. I do have some notion of computer programs having meaning. C, python, C++, ocaml, even Coq mean something to me because they create artifacts that exist and live outside of myself. They run. Their execution becomes fact regardless if I understand them or not. Executable programs become a part of the physical world. I do not feel the same about mathematics or computer science written down in papers. It is static and only moves in the minds of people. I do not feel compelled to consider what is written in a paper as fact, a fait accompli, if I do not personally understand it. I don’t think many people are trying to willfully deceive me in the academic literature, but there is always the possibility and that of self deception.
So for me, a pile of cryptic symbols with horizontal lines is unacceptable and contains almost no meaning. It begins to have meaning when I know how to translate the concepts into an executable programming language.
These concepts also being to have meaning if I personally know how to manipulate them with paper and pencil.
To put symbolic mathematical stuff in a computer, a standard methodology is to consider expressions as trees. We then want to manipulate those trees in programs or produce new trees.
Something that gives me conniptions is that implication \(\rimpl\), turnstile $\vdash$, And horizontal bars are all some kind of notion of logic implication, albeit with the subtlest conceptual and mechanical distinctions. Somehow it the interplay of pushing things around between these three similar things encapsulates a notion of logical deduction. Super bizarre.
B is mangetic field, Gamma is a context
What are types?
Checking is easy? Inference is hard?
Is this always true? Checking of application is hard f x : b because it needs to invent an inner type.
We can make checking always go through with intense annotations.
infertype below the line is good. It is producing information infertype above the line is bad. It needs a information producing process to output
It almost feels like a constraint satisfaction problem to me. Perhaps the more flexible notion of a prolog search with pruning.
We want to pick an ordering and infer/check for the rules such that they are well moded. We want as many infers below lines and checks above lines as possible. These may be weighted by some notion of painfulness of annotation. Freqeuncy of usage * annotation size perhaps. To require a synethsizing derivation is bad. To produce a synthesizing derivation is good.
The Pfenning recipe is a heuristic
Well moded : premise known.
The rules aren’t independent of each other though? How does that hjapen? Yes. The rules need to match up to each other. Maybe really the cost is every annotation. which happens when there is a form that infers up top but checks below the line.
Of course, a language designer is not constrained to a fixed system. Adding extra forms,, or interesintg new contexts or ways of thinkgin about things.
elimination forms are often in inference because there is an invented type
typedirected synthesis of programs is proof search.
 principal judgement
 elimination synethesize, introduction check. Elimination is usually above and introuducion usually below. This is counterintuitive, we have to step backward to step forward. WWe don’t want synthesis above and check below.
A pile of introductions should just check? Carry the top level information through
subtyping :< often has input mode. It’s easy to tell if things are subtypes than to come up with a maximal type persay
We don’t want to dump information
The simply typed lambda calculus is kind of a bad example. It is so simple that really the nonbidirectional version does a pretty dang good job in raw prolog.
Inference rules: What is their computational reading/encoding? The notation is extremely vague if this is your desire.
 The expression below the line defines a relation.
Relations may be represented different ways R a b c Am I mixing types and expressions
 [(a, b, c)]  explicit list (possibly infinite)
 a > b > c > bool (decidable or partial depending on language) indicator function
 Halfsies. a > [(b,c)], a > b > [c], in this sense a > b > c > bool is really a > b > c > [()]
 type Rel a b c = a > b > c > Prop – a formula

explicit data type. data Rel = True False And (Rel) (Rel) Or  R(A,B,C). – prolog
 C C++, Java, Python? Do they give any new ideas?
Ok but what about Rules. Different but related question.
 Prolog clause. Problems
 Coq Inductive data type
 Cases in recursive function definitions
 a pile of switch cases or whatever
 Tactics  th below > ( [ th above ], [Pf above] > Pf below )
 LCF Recognizers  typ rule = [Pf up top] > theore below > Pf below  sometimes yuou need to give the shape of the thing you want below. Even all the above parts and the label of the rule might be insufficient. It is possible for this function to be partial / undecidable. Proof terms are a deep DSL recording these shallow moves.
 You could make an argument for an inference rule being a relation of relations? A meta lifting.
 rules with single things up top = executions of abstract machines
Inference rules are relations on syntactic forms?
This notion of modes is essentially the “Halfsies” notion. Bidirectional typechecking defines the relation not between
A typing judgement is a relation between contexts, expressions, and types. Expressions have types. Values have types. These are distinct notions. Value as a meta notion vs Value as a subclass of expressions.
(tenv, expr, typ ) > true (tenv, expr) > Maybe typ
type var = string type expr = Var of var  Lam of var expr  App of (expr expr)  TT  Annot of (expr typ) type typ = Unit  Arr of typ typ type typ_ctx = (var * typ) list
type_check gamma = function  Var v,  TT , Unit > true  e, t > let t’ = type_infer gamma e in t = t’ and type_infer gamma = function  Var v > lookup gamma v  Annot e t > let c = type_check gamma e t in if c then t else failwith “Bad annotation”  App f x > let Arr (a,b) = infer_type gamma f in let c = type_check gamma x a in if c then b else fail  _ > failwith
Does it matter that we should return errors? I guess it does.
type_judge(Gamma, e, type). type_judge(Gamma, tt, unit). type_judge(Gamma, lam(V,Body) , arr(A,B)) : type_judge([ V > A  Gamma], Body , B ). type_judge(Gamma, app(F,X) , ) :
He mentioned thaty a computational interpretation of inference rules requirtes modes Symntax directed rules can be considered as checkers, since we can pattern match on the syntax and know what to do
What interesting problem can I use? Well, interleaved moded predicates is interesting.
Mode correctness  not a notion of bidirectional type checking.
This is an interesting idea. Take a prolog program that is ill formed. Consider all possible modal forms. In recursiove calls, consider all possible modes to recrusviely call. Consider all orderings of right side preds too. This is all mechanical enough it could be considered mode synthesis. Perhaps + some test cases
rel(A,B) : rel(A,C), rel(C,B).
rel+(A,+B) : rel–, rel–
Trim all non well moded
Maybe I hsould use mercury? Or swipl has a mode chekcing thing probably? https://www.swiprolog.org/pldoc/man?section=modes
The fool.
It has something to do with focusing? Hmm The review paper says explicitly it does not.
https://twitter.com/rob_rix/status/1379966266723680256?s=20 rob rix dicssuion polarity sterling says look at sequent. negative have noninvertible left rules, positive has noninvertible right Substructural logic (like linear) What about bunched logic like seperation? Does that add any clairty? What is the polarity of seperating conjunction?
https://github.com/jonsterling/dreamtt “pedagogical” binrectional dependent tpye implentation
https://astampoulis.github.io/blog/taksimmodesmakam/  huh makam has modes inspired by the bidi survey
https://twitter.com/puffnfresh/status/1377474203444637701?s=20
https://consequently.org/papers/dggl.pdf
https://personal.cis.strath.ac.uk/neil.ghani/papers/ghanijfcs09.pdf
Positive and negative types
Focusing
The two G and D categories in Harrop/Horn formula
Positivityv checking
Covaraince and contravaraince
hmm. Harper seems to suggest a positive v neagtive product type is a lazy vs strict distinction https://ncatlab.org/nlab/show/product+type 
https://ncatlab.org/nlab/show/sum+type  need multiple sequents for
Linear logic.
Positive is “charactersize “ by introductions. Plays nice with eager evaluation Negative is “characterized” by eliminations. Plays “nice” with lazy evaluation
Introduction rules, elimination rules AND computation rules?
Proof search vs Preoof normalization semantics.
focussing and prolog
hmm harper actually has a chapter on classical logic dynamics
Bidrectional typing Type synthesis can be interleaved with type checking
Session types to understand linear logic vs memory management Types are things that can be attached to terms. Terms are expressions Process calculi are some kind of terms. https://homepages.inf.ed.ac.uk/wadler/papers/propositionsassessions/propositionsassessions.pdf
tactics are perhaps another operational interpetation of inference rules https://hackage.haskell.org/package/refinery https://reasonablypolymorphic.com/blog/towardstactics/ https://www.youtube.com/watch?v=S0HvfXq3454&ab_channel=Z%C3%BCrichFriendsofHaskell
Misc
smalltt a demo for high performance type theory elaboration. Really good readme too elboration seminar learn type theory simple observationsl type theory
regular propositions are basically the “constructivecontentfree” parts of propositions
 https://counterexamples.org/title.html Counterexamples in Type Systems. Really cool resources of problems that can show up in type systems
 https://github.com/FStarLang/FStar/issues/360 Examples that show paradoxes when using combinations of impredicative polymorphism + excluded middle + large elimination – can only choose two https://cstheory.stackexchange.com/questions/21836/whydoescoqhaveprop/21878#21878 Berardi’s paradox
https://github.com/coq/coq/issues/9458 future of impredicative set
introductory type theory resources https://www.haskellforall.com/2022/05/introductoryresourcestotypetheory.html
Amusingly, by mixing the type level and value level, do we end up with a situation akin to dynamic types where we often need to carry types a runtime? Or is that mistaken?
Type Theory in Color internalizes parametricity results
Programming = proving? The CurryHoward correspondence today Xavier Leroy  so freakin good
Books
 TAPL
 Programming Language Foundations  Harper
 Programming in Martin Lof Type Theory
 Software Foundations
 PLFA
Resources for Teaching with Formal Methods mostly type thoery based