nLab propositional 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 predicate logic over type theory, propositional equality is the notion of equality which is defined to be a proposition. Propositional equality is most commonly used in set theories like ZFC and ETCS, but it could also be used for definitional equality and conversional equality in some presentations of dependent type theories like Martin-Löf type theory or cubical type theory in place of judgmental equality.

Propositional equality can be contrasted with judgmental equality, where equality is a judgment, and typal equality, where equality is a type.

Note on terminology

Historically in the dependent type theory community, the term propositional equality was used for typal equality. This was because under the principle of propositions as types, one interprets all types in a single-layer type theory as being propositions. However, we choose to make a distinction between propositional equality and typal equality. First, propositional equality as defined in this article is used in the most common foundations of mathematics, such as ZFC and ETCS, and is clearly not a type. Additionally, in some logics over type theory, one can have three distinct notions of equality: judgmental equality, propositional equality, and typal equality. Finally, in the advent of homotopy type theory and other type theoretic higher foundations, typal equality is no longer required to be a subsingleton or h-proposition, and the alternative principle of propositions as some types has become the primary interpretation of dependent type theory, where only the subsingletons or h-propositions are interpreted as propositions.

Definition and structural rules

Propositional equality of types and terms is formed by the following rules:

ΓAtypeΓBtypeΓABprop\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \equiv B \; \mathrm{prop}}
ΓAtypeΓa:AΓb:AΓa Abprop\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A}{\Gamma \vdash a \equiv_A b \; \mathrm{prop}}

Propositional equality has its own structural rules: reflexivity, symmetry, transitivity, the principle of substitution, and the variable conversion rule.

  • Reflexivity of propositional equality
ΓAtypeΓAAtrue\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash A \equiv A \; \mathrm{true}}
ΓAtypeΓa:AΓa Aatrue\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A}{\Gamma \vdash a \equiv_A a \; \mathrm{true}}
  • Symmetry of propositional equality
ΓAtypeΓBtypeΓABtrueΓBAtrue\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash A \equiv B \; \mathrm{true}}{\Gamma \vdash B \equiv A \; \mathrm{true}}
ΓAtypeΓa:AΓb:AΓa AbtrueΓb Aatrue\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash a \equiv_A b \; \mathrm{true}}{\Gamma \vdash b \equiv_A a \; \mathrm{true}}
  • Transitivity of propositional equality
ΓAtypeΓBtypeΓCtypeΓABtrueΓBCtrueΓACtrue\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash A \equiv B \; \mathrm{true} \quad \Gamma \vdash B \equiv C \; \mathrm{true}}{\Gamma \vdash A \equiv C \; \mathrm{true}}
ΓAtypeΓa:AΓb:AΓc:AΓa AbtrueΓb ActrueΓa Actrue\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash c:A \quad \Gamma \vdash a \equiv_A b \; \mathrm{true} \quad \Gamma \vdash b \equiv_A c \; \mathrm{true}}{\Gamma \vdash a \equiv_A c \; \mathrm{true}}
  • Principle of substitution of propositionally equal terms:

    ΓAtypeΓa:AΓb:AΓa AbtrueΓ,x:A,ΔBtypeΓ,Δ[b/x]B[a/x]B[b/x]true\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a : A \quad \Gamma \vdash b : A \quad \Gamma \vdash a \equiv_A b \; \mathrm{true} \quad \Gamma, x:A, \Delta \vdash B \; \mathrm{type}}{\Gamma, \Delta[b/x] \vdash B[a/x] \equiv B[b/x] \; \mathrm{true}}
    ΓAtypeΓa:AΓb:AΓa AbtrueΓ,x:A,Δc:BΓ,Δ[b/x]c[a/x] B[b/x]c[b/x]true\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a : A \quad \Gamma \vdash b : A \quad \Gamma \vdash a \equiv_A b \; \mathrm{true} \quad \Gamma, x:A, \Delta \vdash c:B}{\Gamma, \Delta[b/x] \vdash c[a/x] \equiv_{B[b/x]} c[b/x] \; \mathrm{true}}
  • The variable conversion rule for propositionally equal types:

ΓAtypeΓBtypeΓABtrueΓ,x:A,Δ𝒥Γ,x:B,Δ𝒥\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash A \equiv B \; \mathrm{true} \quad \Gamma, x:A, \Delta \vdash \mathcal{J}}{\Gamma, x:B, \Delta \vdash \mathcal{J}}

In addition, if the dependent type theory has type definition judgments BAtypeB \coloneqq A \; \mathrm{type} and term definition judgments ba:Ab \coloneqq a:A, then propositional equality is used in the following rules:

  • Formation and propositional equality reflection rules for type definition:

    ΓBAtypeΓBtypeΓBAtypeΓBAtrue\frac{\Gamma \vdash B \coloneqq A \; \mathrm{type}}{\Gamma \vdash B \; \mathrm{type}} \qquad \frac{\Gamma \vdash B \coloneqq A \; \mathrm{type}}{\Gamma \vdash B \equiv A\; \mathrm{true}}
  • Introduction and propositional equality reflection rules for term definition:

    Γba:AΓb:AΓba:AΓb Aatrue\frac{\Gamma \vdash b \coloneqq a:A}{\Gamma \vdash b:A} \qquad \frac{\Gamma \vdash b \coloneqq a:A}{\Gamma \vdash b \equiv_A a \; \mathrm{true}}

In computation and uniqueness rules

Propositional equality can be used in the computation rules and uniqueness rules of types in dependent type theory:

  • Computation rules for dependent product types:
Γ,x:Ab(x):B(x)Γa:AΓλ(x:A).b(x)(a) B[a/x]b[a/x]true\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/x]} b[a/x] \; \mathrm{true}}
  • Uniqueness rules for dependent product types:
Γf: x:AB(x)Γf x:AB(x)λ(x).f(x)true\frac{\Gamma \vdash f:\prod_{x:A} B(x)}{\Gamma \vdash f \equiv_{\prod_{x:A} B(x)} \lambda(x).f(x) \; \mathrm{true}}
  • Computation rules for dependent sum types:
Γ,x:Ab(x):B(x)Γa:AΓπ 1(a,b) AatrueΓ,x:Ab:BΓa:AΓπ 2(a,b) B(π 1(a,b))btrue\frac{\Gamma, x:A \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \pi_1(a, b) \equiv_A a \; \mathrm{true}} \qquad \frac{\Gamma, x:A \vdash b:B \quad \Gamma \vdash a:A}{\Gamma \vdash \pi_2(a, b) \equiv_{B(\pi_1(a, b))} b \; \mathrm{true}}
  • Uniqueness rules for dependent sum types:
Γz: x:AB(x)Γz x:AB(x)(π 1(z),π 2(z))true\frac{\Gamma \vdash z:\sum_{x:A} B(x)}{\Gamma \vdash z \equiv_{\sum_{x:A} B(x)} (\pi_1(z), \pi_2(z)) \; \mathrm{true}}
  • Computation rules for identity types:
Γ,a:A,b:A,p:a= AbCtypeΓ,c:At:C[c/a,c/b,refl A(c)/p]Γ,c:AJ(t,c,c,refl(c)) C[c/a,c/b,refl A(c)/p]ttrue\frac{\Gamma, a:A, b:A, p:a =_A b \vdash C \; \mathrm{type} \quad \Gamma, c:A \vdash t:C[c/a, c/b, \mathrm{refl}_A(c)/p]}{\Gamma, c:A \vdash J(t, c, c, \mathrm{refl}(c)) \equiv_{C[c/a, c/b, \mathrm{refl}_A(c)/p]} t \; \mathrm{true}}

 See also

Last revised on December 16, 2022 at 16:17:52. See the history of this page for a list of all contributions to it.