nLab judgmental equality


Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Equality and Equivalence



In any type theory, judgmental equality is the notion of equality which is defined to be a judgment. Judgmental equality is most commonly used in single-level type theories like Martin-Löf type theory or higher observational type theory for making inductive definitions, but it is also used in cubical type theory and simplicial type theory to define probe shapes for (infinity,1)-categorical types which could not be coherently defined in vanilla dependent type theory.

In general, there are two different kinds of judgmental equalities

  • strict judgmental equalities, where judgmental equalities behave like the strict equality of a set and is preserved throughout the type theory as congruences

  • weak judgmental equalities, where judgmental equalities are judgmental representations for identifications between terms of a type or equivalences of types; i.e. where ab:Aa \equiv b:A is formally a shorthand for p:a= Abp:a =_A b and similarly ABA \equiv B is shorthand for e:ABe:A \simeq B

Strict judgmental equalities are used in most dependent type theories. Weak judgmental equalities can be used in weak type theories, where a direct translation of the inference rules of the types in Martin-Löf type theory results in a weak version of Martin-Löf type theory.

Judgmental equality can be contrasted with propositional equality, where equality is a proposition, and typal equality, where equality is a type.


In the model of dependent type theory which uses judgmental equality for definitional equality, judgmental equality of types, terms, and contexts are given by the following judgments:

  • ΓAAtype\Gamma \vdash A \equiv A' \; \mathrm{type} - AA and AA' are judgementally equal well-typed types in context Γ\Gamma.
  • Γaa:A\Gamma \vdash a \equiv a' : A - aa and aa' are judgementally equal well-typed terms of type AA in context Γ\Gamma.
  • ΓΓctx\Gamma \equiv \Gamma' \; \mathrm{ctx} - Γ\Gamma and Γ\Gamma' are judgementally equal contexts.

Inference rules for judgmental equality

Weak judgmental equality

Weak judgmental equality of terms is simply given by a reflection rule into the identity type:

Γaa:AΓδ a,a:a= Aa\frac{\Gamma \vdash a \equiv a':A}{\Gamma \vdash \delta_{a, a'}:a =_A a'}

Weak judgmental equality of types is given by one of the two sets of structural rules:

  • The variable conversion rule for judgmentally equal types:
    ΓABtypeΓ,x:A,Δ𝒥Γ,x:B,Δ𝒥\frac{\Gamma \vdash A \equiv B \; \mathrm{type} \quad \Gamma, x:A, \Delta \vdash \mathcal{J}}{\Gamma, x:B, \Delta \vdash \mathcal{J}}


  • Rules for isomorphisms between judgmentally equal types:
ΓABtypeΓ,x:Aδ A,B(x):BΓABtypeΓ,y:Bδ A,B 1(x):A\frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma, x:A \vdash \delta_{A, B}(x):B} \qquad \frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma, y:B \vdash \delta_{A, B}^{-1}(x):A}
ΓABtypeΓ,x:Aδ A,B 1(δ A,B(x))x:AΓABtypeΓ,y:Bδ A,B(δ A,B 1(y))y:B\frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma, x:A \vdash \delta_{A, B}^{-1}(\delta_{A, B}(x)) \equiv x:A} \qquad \frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma, y:B \vdash \delta_{A, B}(\delta_{A, B}^{-1}(y)) \equiv y:B}

In the first case, one could construct isomorphisms from the variable conversion rule, other structural rules, and the rules for function types:

From the generic term rule and the variable conversion rule for judgmentally equal types AAA \equiv A' we have x:Ax:Ax:A' \vdash x:A and x:Ax:Ax:A \vdash x:A', whereby from the introduction and computation rules for function types we have functions λx:A.x:AA\lambda x:A'.x:A' \to A and λx:A.x:AA\lambda x:A.x:A \to A' such that

(λx:A.x)((λx:A.x)(x))(λx:A.x)(x)x:A(\lambda x:A'.x)((\lambda x:A.x)(x)) \equiv (\lambda x:A'.x)(x) \equiv x:A
(λx:A.x)((λx:A.x)(x))(λx:A.x)(x)x:A(\lambda x:A.x)((\lambda x:A'.x)(x)) \equiv (\lambda x:A.x)(x) \equiv x:A'

making both functions λx:A.x\lambda x:A'.x and λx:A.x\lambda x:A.x isomorphisms.

Strict judgmental equality

Strict judgmental equality of terms has congruence rules for substitution, the principle of substitution:

  • Principle of substitution for judgmentally equal terms:
    ΓAtypeΓab:AΓ,x:A,ΔB(x)typeΓ,Δ(a)B(a)B(b)type\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash B(x) \; \mathrm{type}}{\Gamma, \Delta(a) \vdash B(a) \equiv B(b) \; \mathrm{type}}
    ΓAtypeΓab:AΓ,x:A,Δc(x):B(x)Γ,Δ(a)c(a)c(b):B(a)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash c(x):B(x)}{\Gamma, \Delta(a) \vdash c(a) \equiv c(b): B(a)}

This implies the reflection rule of weak judgmental equalities because one could derive the following rule:

Γab:AΓrefl A(a):a= Ab\frac{\Gamma \vdash a \equiv b:A}{\Gamma \vdash \mathrm{refl}_A(a):a =_A b}

In addition to the variable conversion rule, there are reflexivity, symmetry, and transitivity rules making strict judgmental equality for both terms and types an equivalence relation:

  • Reflexivity of judgmental equality
ΓAtypeΓAAtype\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash A \equiv A \; \mathrm{type}}
ΓAtypeΓa:AΓaa:A\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A}{\Gamma \vdash a \equiv a:A}
ΓctxΓΓctx\frac{\Gamma \; \mathrm{ctx}}{\Gamma \equiv \Gamma \; \mathrm{ctx}}
  • Symmetry of judgmental equality

    ΓABtypeΓBAtype\frac{\Gamma \vdash A \equiv B \; \mathrm{type}}{\Gamma \vdash B \equiv A \; \mathrm{type}}
    ΓAtypeΓab:AΓba:A\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b:A}{\Gamma \vdash b \equiv a:A}
    ΓΔctxΔΓctx\frac{\Gamma \equiv \Delta \; \mathrm{ctx}}{\Delta \equiv \Gamma \; \mathrm{ctx}}
  • Transitivity of judgmental equality

    ΓABtypeΓBCtypeΓACtype\frac{\Gamma \vdash A \equiv B \; \mathrm{type} \quad \Gamma \vdash B \equiv C \; \mathrm{type} }{\Gamma \vdash A \equiv C \; \mathrm{type}}
    ΓAtypeΓab:Abc:AΓac:A\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a \equiv b:A \quad b \equiv c:A }{\Gamma \vdash a \equiv c:A}
    ΓΔctxΔΞctxΓΞctx\frac{\Gamma \equiv \Delta \; \mathrm{ctx} \quad \Delta \equiv \Xi \; \mathrm{ctx}}{\Gamma \equiv \Xi \; \mathrm{ctx}}

Congruence rules for judgmental equality

In addition, strict judgmental equalities have congruence rules for every type in the type theory.

  • Congruence rules for dependent function types
ΓAAtypeΓ,x:AB(x)B(x)typeΓ x:AB(x) x:AB(x)type\frac{ \begin{array}{c} \Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \end{array} }{\Gamma \vdash \prod_{x:A} B(x) \equiv \prod_{x:A'} B'(x)\; \mathrm{type}}
ΓAtypeΓ,x:AB(x)typeΓ,x:Ab(x):B(x)Γ,x:Ab(x):B(x) Γ,x:Ab(x)b(x):B(x)Γλx:A.b(x)λx:A.b(x): x:A.B(x)\frac{ \begin{array}{c} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x) \quad \Gamma, x:A \vdash b'(x):B(x) \\ \Gamma, x:A \vdash b(x) \equiv b'(x):B(x) \end{array} }{\Gamma \vdash \lambda x:A.b(x) \equiv \lambda x:A.b'(x):\prod_{x:A}.B(x)}
ΓAtypeΓ,x:AB(x)typeΓf: x:AB(x)f: x:AB(x) Γff: x:AB(x)Γ,x:Af(x)f(x):B(x)\frac{ \begin{array}{c} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B(x) \quad f':\prod_{x:A} B(x) \\ \Gamma \vdash f \equiv f':\prod_{x:A} B(x) \end{array} }{\Gamma, x:A \vdash f(x) \equiv f'(x):B(x)}
ΓAtypeΓ,x:AB(x)typeΓ,x:Ab(x):B(x)Γ,x:Ab(x):B(x) Γ,x:Ab(x)b(x):B(x)Γβ A,Bx:A.b(x)β A,Bx:A.b(x): x:Ab(x)= B(x)(λx:A.b(x))(x)\frac{ \begin{array}{c} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x) \quad \Gamma, x:A \vdash b'(x):B(x) \\ \Gamma, x:A \vdash b(x) \equiv b'(x):B(x) \end{array} }{\Gamma \vdash \beta_{\prod}^{A, B} x:A.b(x) \equiv \beta_{\prod}^{A, B} x:A.b'(x):\prod_{x:A} b(x) =_{B(x)} (\lambda x:A.b(x))(x)}
ΓAtypeΓ,x:AB(x)typeΓ,x:AB(x)type Γ,x:AB(x)B(x)typeΓη A,Bη A,B: f: x:AB(x)f= x:AB(x)λx:A.f(x)\frac{ \begin{array}{c} \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash B'(x) \; \mathrm{type} \\ \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \end{array} }{\Gamma \vdash \eta_{\prod}^{A, B} \equiv \eta_{\prod}^{A, B'}:\prod_{f:\prod_{x:A} B(x)} f =_{\prod_{x:A} B(x)} \lambda x:A.f(x)}
  • Congruence rules for dependent pair types:
ΓAAtypeΓ,x:AB(x)B(x)typeΓ x:AB(x) x:AB(x)type\frac{ \begin{array}{c} \Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \end{array} }{\Gamma \vdash \sum_{x:A} B(x) \equiv \sum_{x:A'} B'(x)\; \mathrm{type}}
ΓAAtypeΓ,x:AB(x)B(x)typeΓ,x:A,y:B(x)pair A,Bpair A,B: x:AB(x)\frac{ \begin{array}{c} \Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \end{array} }{\Gamma, x:A, y:B(x) \vdash \mathrm{pair}_{\sum}^{A, B} \equiv \mathrm{pair}_{\sum}^{A', B'}:\sum_{x:A} B(x)}
ΓAAtypeΓ,x:AB(x)B(x)typeΓ,z: x:AB(x)C(z)C(z)typeΓind A,B,Cind A,B,C: g: x:A y:B(x)C(pair A,B(x,y)) z: x:AB(x)C(z)\frac{ \begin{array}{c} \Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \quad \Gamma, z:\sum_{x:A} B(x) \vdash C(z) \equiv C'(z) \; \mathrm{type} \end{array} }{\Gamma \vdash \mathrm{ind}_{\sum}^{A, B, C} \equiv \mathrm{ind}_{\sum}^{A', B', C'}:\prod_{g:\prod_{x:A} \prod_{y:B(x)} C(\mathrm{pair}_{\sum}^{A, B}(x, y))} \prod_{z:\sum_{x:A} B(x)} C(z)}
ΓAAtypeΓ,x:AB(x)B(x)typeΓ,z: x:AB(x)C(z)C(z)typeΓβ A,B,Cβ A,B,C: g: x:A y:B(x)C(pair A,B(x,y)) x:A y:B(x)ind A,B,C(g,pair A,B(x,y))= C(pair A,B(x,y))g(x,y)\frac{ \begin{array}{c} \Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \quad \Gamma, z:\sum_{x:A} B(x) \vdash C(z) \equiv C'(z) \; \mathrm{type} \end{array} }{\Gamma \vdash \beta_{\sum}^{A, B, C} \equiv \beta_{\sum}^{A', B', C'}:\prod_{g:\prod_{x:A} \prod_{y:B(x)} C(\mathrm{pair}_{\sum}^{A, B}(x, y))} \prod_{x:A} \prod_{y:B(x)} \mathrm{ind}_{\sum}^{A, B, C}(g, \mathrm{pair}_{\sum}^{A, B}(x, y)) =_{C(\mathrm{pair}_{\sum}^{A, B}(x, y))} g(x, y)}
  • Congruence rules for identity types:
ΓAAtypeΓ,x:A,y:Ax= Ayx= Ay\frac{\Gamma \vdash A \equiv A' \; \mathrm{type}}{\Gamma, x:A, y:A \vdash x =_A y \equiv x =_{A'} y}
ΓAAtypeΓrefl Arefl A: x:Ax= Ax\frac{\Gamma \vdash A \equiv A' \; \mathrm{type}}{\Gamma \vdash \mathrm{refl}_A \equiv \mathrm{refl}_{A'}:\prod_{x:A} x =_A x}
ΓAAtypeΓ,x:A,y:A,p:x= AyC(x,y,p)C(x,y,p)typeΓind = A,Cind = A,C: t: x:AC(x,x,refl A(x)) x:A y:A p:x= AyC(x,y,p)\frac{ \begin{array}{c} \Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A, y:A, p:x =_A y \vdash C(x, y, p) \equiv C'(x, y, p) \; \mathrm{type} \end{array} }{\Gamma \vdash \mathrm{ind}_{=}^{A, C} \equiv \mathrm{ind}_{=}^{A', C'}:\prod_{t:\prod_{x:A} C(x, x, \mathrm{refl}_A(x))} \prod_{x:A} \prod_{y:A} \prod_{p:x =_A y} C(x, y, p)}
ΓAAtypeΓ,x:A,y:A,p:x= AyC(x,y,p)C(x,y,p)typeΓβ ind = A,Cβ ind = A,C: t: x:AC(x,x,refl A(x)) x:Aind = A,C(t,x,x,refl A(x))= C(x,x,refl A(x))t(x)\frac{ \begin{array}{c} \Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A, y:A, p:x =_A y \vdash C(x, y, p) \equiv C'(x, y, p) \; \mathrm{type} \end{array} }{\Gamma \vdash \beta_{ \mathrm{ind}_=}^{A, C} \equiv \beta_{\mathrm{ind}_=}^{A', C'}:\prod_{t:\prod_{x:A} C(x, x, \mathrm{refl}_A(x))} \prod_{x:A} \mathrm{ind}_{=}^{A, C}(t, x, x, \mathrm{refl}_A(x)) =_{C(x, x, \mathrm{refl}_A(x))} t(x)}
  • Congruence rules for the empty type:
Γ,x:C(x)C(x)typeΓind Cind C: x:C(x)type\frac{\Gamma, x:\emptyset \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_\emptyset^C \equiv \mathrm{ind}_\emptyset^{C'}:\prod_{x:\emptyset} C(x) \; \mathrm{type}}
  • Congruence rules for the type of booleans:
Γ,x:𝟚C(x)C(x)typeΓind 𝟚 Cind 𝟚 C: a:C(0) b:C(1) x:𝟚C(x)\frac{\Gamma, x:\mathbb{2} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_\mathbb{2}^C \equiv \mathrm{ind}_\mathbb{2}^{C'}:\prod_{a:C(0)} \prod_{b:C(1)} \prod_{x:\mathbb{2}} C(x)}
Γ,x:𝟚C(x)C(x)typeΓβ 𝟚 0,Cβ 𝟚 0,C: a:C(0) b:C(1)ind 𝟚 C(a,b,0)= C(0)a\frac{\Gamma, x:\mathbb{2} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{2}^{0, C} \equiv \beta_\mathbb{2}^{0, C'}:\prod_{a:C(0)} \prod_{b:C(1)} \mathrm{ind}_\mathbb{2}^C(a, b, 0) =_{C(0)} a}
Γ,x:𝟚C(x)C(x)typeΓβ 𝟚 1,Cβ 𝟚 1,C: a:C(0) b:C(1)ind 𝟚 C(a,b,1)= C(1)b\frac{\Gamma, x:\mathbb{2} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{2}^{1, C} \equiv \beta_\mathbb{2}^{1, C'}:\prod_{a:C(0)} \prod_{b:C(1)} \mathrm{ind}_\mathbb{2}^C(a, b, 1) =_{C(1)} b}
  • Congruence rules for the natural numbers type:
Γ,x:C(x)C(x)typeΓind Cind C: c 0:C(0) c s: x:C(x)C(s(x)) x:C(x)\frac{\Gamma, x:\mathbb{N} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^C \equiv \mathrm{ind}_\mathbb{N}^{C'}:\prod_{c_0:C(0)} \prod_{c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))} \prod_{x:\mathbb{N} C(x)}}
Γ,x:C(x)C(x)typeΓβ 0,Cβ 0,C: c 0:C(0) c s: x:C(x)C(s(x))ind C(c 0,c s,0)= C(0)c 0\frac{\Gamma, x:\mathbb{N} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{N}^{0, C} \equiv \beta_\mathbb{N}^{0, C'}:\prod_{c_0:C(0)} \prod_{c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))} \mathrm{ind}_\mathbb{N}^C(c_0, c_s, 0) =_{C(0)} c_0}
Γ,x:C(x)C(x)typeΓβ s,Cβ s,C: c 0:C(0) c s: x:C(x)C(s(x)) x:ind C(c 0,c s,s(x))= C(s(x))c s(x)(ind C(c 0,c s,x))\frac{\Gamma, x:\mathbb{N} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{N}^{s, C} \equiv \beta_\mathbb{N}^{s, C'}:\prod_{c_0:C(0)} \prod_{c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))} \prod_{x:\mathbb{N}} \mathrm{ind}_\mathbb{N}^C(c_0, c_s, s(x)) =_{C(s(x))} c_s(x)(\mathrm{ind}_\mathbb{N}^C(c_0, c_s, x))}

Similarly, we have congruence rules for every axiom in the dependent type theory, such as

ΓAAtypeΓ,x:AB(x)B(x)typeΓfunext A,Bfunext A,B: f; x:AB(x) g: x:AB(x)(f= x:AB(x)g) x:Af(x)= B(x)g(x)\frac{\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{funext}_{A, B} \equiv \mathrm{funext}_{A', B'}:\prod_{f;\prod_{x:A} B(x)} \prod_{g:\prod_{x:A} B(x)} (f =_{\prod_{x:A} B(x)} g) \simeq \prod_{x:A} f(x) =_{B(x)} g(x)}
ΓAAtypeΓ,x:AB(x)B(x)typeΓ,x:A,y:B(x)C(x,y)C(x,y)typeΓchoice A,B,Cchoice A,B,C:(isSet(A)× x:AisSet(B(x)))x:A.y:B(x).C(x,y)g: x:AB(x).x:A.C(x,g(x))\frac{\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \quad \Gamma, x:A, y:B(x) \vdash C(x, y) \equiv C'(x, y) \; \mathrm{type}}{\Gamma \vdash \mathrm{choice}_{A, B, C} \equiv \mathrm{choice}_{A', B', C'}:\left(\mathrm{isSet}(A) \times \prod_{x:A} \mathrm{isSet}(B(x))\right) \to \forall x:A.\exists y:B(x).C(x, y) \to \exists g:\prod_{x:A} B(x).\forall x:A.C(x, g(x))}

In computation and uniqueness rules

Judgmental equality of terms can be used in the computation rules and uniqueness rules of types:

  • Computation rules for dependent product types:
Γ,x:Ab(x):B(x)Γa:AΓλ(x:A).b(x)(a)b(a):B(a)\frac{\Gamma, x:A \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \lambda(x:A).b(x)(a) \equiv b(a):B(a)}
  • Uniqueness rules for dependent product types:
Γf: x:AB(x)Γfλ(x).f(x): x:AB(x)\frac{\Gamma \vdash f:\prod_{x:A} B(x)}{\Gamma \vdash f \equiv \lambda(x).f(x):\prod_{x:A} B(x)}
  • Computation rules for negative dependent sum types:
Γ,x:Ab(x):B(x)Γa:AΓπ 1(a,b)a:AΓ,x:Ab:BΓa:AΓπ 2(a,b)b:B(π 1(a,b))\frac{\Gamma, x:A \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \pi_1(a, b) \equiv a:A} \qquad \frac{\Gamma, x:A \vdash b:B \quad \Gamma \vdash a:A}{\Gamma \vdash \pi_2(a, b) \equiv b:B(\pi_1(a, b))}
  • Uniqueness rules for negative dependent sum types:
Γz: x:AB(x)Γz(π 1(z),π 2(z)): x:AB(x)\frac{\Gamma \vdash z:\sum_{x:A} B(x)}{\Gamma \vdash z \equiv (\pi_1(z), \pi_2(z)):\sum_{x:A} B(x)}
  • Computation rules for identity types:
    Γ,a:A,b:A,p:a= AbC(a,b,p)typeΓt: c:AC(c,c,refl A(c))Γ,c:AJ(t,c,c,refl(c))t:C(c,c,refl A(c))\frac{\Gamma, a:A, b:A, p:a =_A b \vdash C(a, b, p) \; \mathrm{type} \quad \Gamma \vdash t:\prod_{c:A} C(c, c, \mathrm{refl}_A(c))}{\Gamma, c:A \vdash J(t, c, c, \mathrm{refl}(c)) \equiv t:C(c, c, \mathrm{refl}_A(c))}

See also


  • Robin Adams, Pure type systems with judgemental equality, Journal of Functional Programming, Volume 16 Issue 2(2006) (web)

  • Vincent Siles, Hugo Herbelin, Equality is typable in semi-full pure type systems (pdf)

  • Egbert Rijke, Introduction to Homotopy Type Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press (pdf) (478 pages)

Last revised on October 27, 2023 at 18:24:04. See the history of this page for a list of all contributions to it.