nLab type of entire relations

Redirected from "type of all entire relations".

Context

Universes

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
propositionsetobjecttype
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

semantics

Contents

Idea

In predicative constructive set theory, there is a hierarchy of possible axioms, starting from the existence of function sets and ending with the existence of power sets representing impredicativity. Intermediate is the existence of the set of entire relations, which is given by the axiom of fullness in set theory.

In predicative constructive dependent type theory, there is a similar hierarchy of possible axioms, which begins with the existence of dependent product types (corresponding to function sets in set theory) and ends with the existence of both dependent product types and the type of all propositions (corresponding to power sets in set theory), representing impredicativity. Intermediate to both that is the existence of the type of all entire relations between two types AA and BB, which is the type theoretic equivalent of the axiom of fullness.

Types of entire relations are important in dependent type theory because, assuming propositional truncations, quotient sets, and natural numbers, they allow the user to define the type of multivalued Cauchy real numbers - which are equivalent to Dedekind real numbers - predicatively (i.e. without power sets).

Definition

Recall that an entire relation between types AA and BB in type theory is a family x:A,y:BR(x,y)x:A, y:B \vdash R(x, y) such that for all x:Ax:A

  • for all y:Ay:A, R(x,y)R(x, y) is an h-proposition, and

  • there exists an y:Ay:A such that R(x,y)R(x, y).

isEntire(x:A.y:B.R(x,y)) x:A( u:AisProp(R(x,y)))×y:A.R(x,y)\mathrm{isEntire}(x:A.y:B.R(x, y)) \equiv \prod_{x:A} \left(\prod_{u:A} \mathrm{isProp}(R(x, y))\right) \times \exists y:A.R(x, y)

If there exists a type of propositions Prop\mathrm{Prop}, then the type of entire relations is given by the type

R:A×BPropisEntire(R)\sum_{R:A \times B \to \mathrm{Prop}} \mathrm{isEntire}(R)

where

isEntire(R) x:Ay:B.R(x,y)\mathrm{isEntire}(R) \equiv \prod_{x:A} \exists y:B.R(x, y)

for Russell types of propositions and

isEntire(R) x:Ay:B.El(R(x,y))\mathrm{isEntire}(R) \equiv \prod_{x:A} \exists y:B.\mathrm{El}(R(x, y))

for Tarski types of propositions. Otherwise, we could use inference rules to directly define the type of all entire relations between types AA and BB:

A la Russell

A la Russell, the type of all entire relations is given by the following rules:

Formation rules:

ΓAtypeΓBtypeΓEntRel(A,B)type\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{EntRel}(A, B) \; \mathrm{type}}

Introduction rules:

ΓAtypeΓBtypeΓ,x:A,y:BR(x,y)ΓtoElem A,B,R:isEntire(x:A.y:B.R(x,y))EntRel(A,B)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y)}{\Gamma \vdash \mathrm{toElem}_{A, B, R}:\mathrm{isEntire}(x:A.y:B.R(x, y)) \to \mathrm{EntRel}(A, B)}

Elimination rules:

ΓAtypeΓBtypeΓR:EntRel(A,B)Γ,x:A,y:BR(x,y)type\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma, x:A, y:B \vdash R(x, y) \; \mathrm{type}}
ΓAtypeΓBtypeΓR:EntRel(A,B)Γentrelwitn(R):isEntire(x:A.y:B.R(x,y))\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{entrelwitn}(R):\mathrm{isEntire}(x:A.y:B.R(x, y))}

Computation rules:

ΓAtypeΓBtypeΓ,x:A,y:BR(x,y)Γp:isEntire(x:A.y:B.R(x,y))Γ,x:A,y:B(toElem A,B,R(p))(x,y)R(x,y)type\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma, x:A, y:B \vdash (\mathrm{toElem}_{A, B, R}(p))(x, y) \equiv R(x, y) \; \mathrm{type}}
  • Judgmental computation rules
ΓAtypeΓBtypeΓ,x:A,y:BR(x,y)Γp:isEntire(x:A.y:B.R(x,y))Γentrelwitn(toElem A,B,R(p))p:isEntire(x:A.y:B.R(x,y))\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma \vdash \mathrm{entrelwitn}(\mathrm{toElem}_{A, B, R}(p)) \equiv p:\mathrm{isEntire}(x:A.y:B.R(x, y))}
  • Typal computation rules
ΓAtypeΓBtypeΓ,x:A,y:BR(x,y)Γp:isEntire(x:A.y:B.R(x,y))Γβ EntRel(A,B) entrelwitn,R:entrelwitn(toElem A,B,R(p))= isEntire(x:A.y:B.R(x,y))p\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma \vdash \beta_{\mathrm{EntRel}(A, B)}^{\mathrm{entrelwitn}, R}:\mathrm{entrelwitn}(\mathrm{toElem}_{A, B, R}(p)) =_{\mathrm{isEntire}(x:A.y:B.R(x, y))} p}

Uniqueness rules:

  • Judgmental uniqueness rules:
ΓAtypeΓBtypeΓR:EntRel(A,B)ΓtoElem A,B,R(entrelwitn(R))R:EntRel(A,B)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{toElem}_{A, B, R}(\mathrm{entrelwitn}(R)) \equiv R:\mathrm{EntRel}(A, B)}
  • Typal uniqueness rules:
ΓAtypeΓBtypeΓR:EntRel(A,B)Γη EntRel(A,B)(R):toElem A,B,R(entrelwitn(R))= EntRel(A,B)R\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \eta_{\mathrm{EntRel}(A, B)}(R):\mathrm{toElem}_{A, B, R}(\mathrm{entrelwitn}(R)) =_{\mathrm{EntRel}(A, B)} R}

Extensionality rule:

ΓAtypeΓBtypeΓR:EntRel(A,B)ΓS:EntRel(A,B)Γentrelext A,B(R,S):(R= EntRel(A,B)S) x:A y:BR(x,y)S(x,y)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B) \quad \Gamma \vdash S:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{entrelext}_{A, B}(R, S):(R =_{\mathrm{EntRel}(A, B)} S) \simeq \prod_{x:A} \prod_{y:B} R(x, y) \simeq S(x, y)}

Strictly a la Tarski

Strictly a la Tarski, the type of all entire relations is given by the following rules:

Formation rules:

ΓAtypeΓBtypeΓEntRel(A,B)type\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{EntRel}(A, B) \; \mathrm{type}}

Introduction rules:

ΓAtypeΓBtypeΓ,x:A,y:BR(x,y)ΓtoElem A,B,R:isEntire(x:A.y:B.R(x,y))EntRel(A,B)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y)}{\Gamma \vdash \mathrm{toElem}_{A, B, R}:\mathrm{isEntire}(x:A.y:B.R(x, y)) \to \mathrm{EntRel}(A, B)}

Elimination rules:

ΓAtypeΓBtypeΓR:EntRel(A,B)Γ,x:A,y:BEl(R,x,y)type\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma, x:A, y:B \vdash \mathrm{El}(R, x, y) \; \mathrm{type}}
ΓAtypeΓBtypeΓR:EntRel(A,B)Γentrelwitn(R):isEntire(x:A.y:B.El(R,x,y))\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{entrelwitn}(R):\mathrm{isEntire}(x:A.y:B.\mathrm{El}(R, x, y))}

Computation rules:

ΓAtypeΓBtypeΓ,x:A,y:BR(x,y)Γp:isEntire(x:A.y:B.R(x,y))Γ,x:A,y:BEl(toElem A,B,R(p),x,y)R(x,y)type\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma, x:A, y:B \vdash \mathrm{El}(\mathrm{toElem}_{A, B, R}(p), x, y) \equiv R(x, y) \; \mathrm{type}}
  • Judgmental computation rules
ΓAtypeΓBtypeΓ,x:A,y:BR(x,y)Γp:isEntire(x:A.y:B.R(x,y))Γentrelwitn(toElem A,B,R(p))p:isEntire(x:A.y:B.R(x,y))\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma \vdash \mathrm{entrelwitn}(\mathrm{toElem}_{A, B, R}(p)) \equiv p:\mathrm{isEntire}(x:A.y:B.R(x, y))}
  • Typal computation rules
ΓAtypeΓBtypeΓ,x:A,y:BR(x,y)Γp:isEntire(x:A.y:B.R(x,y))Γβ EntRel(A,B) entrelwitn,R:entrelwitn(toElem A,B,R(p))= isEntire(x:A.y:B.R(x,y))p\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma \vdash \beta_{\mathrm{EntRel}(A, B)}^{\mathrm{entrelwitn}, R}:\mathrm{entrelwitn}(\mathrm{toElem}_{A, B, R}(p)) =_{\mathrm{isEntire}(x:A.y:B.R(x, y))} p}

Uniqueness rules:

  • Judgmental uniqueness rules:
ΓAtypeΓBtypeΓR:EntRel(A,B)ΓtoElem A,B,El(R)(entrelwitn(R))R:EntRel(A,B)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{toElem}_{A, B, \mathrm{El}(R)}(\mathrm{entrelwitn}(R)) \equiv R:\mathrm{EntRel}(A, B)}
  • Typal uniqueness rules:
ΓAtypeΓBtypeΓR:EntRel(A,B)Γη EntRel(A,B)(R):toElem A,B,El(R)(entrelwitn(R))= EntRel(A,B)R\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \eta_{\mathrm{EntRel}(A, B)}(R):\mathrm{toElem}_{A, B, \mathrm{El}(R)}(\mathrm{entrelwitn}(R)) =_{\mathrm{EntRel}(A, B)} R}

Extensionality rule:

ΓAtypeΓBtypeΓR:EntRel(A,B)ΓS:EntRel(A,B)Γentrelext A,B(R,S):(R= EntRel(A,B)S) x:A y:BEl(R,x,y)El(S,x,y)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B) \quad \Gamma \vdash S:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{entrelext}_{A, B}(R, S):(R =_{\mathrm{EntRel}(A, B)} S) \simeq \prod_{x:A} \prod_{y:B} \mathrm{El}(R, x, y) \simeq \mathrm{El}(S, x, y)}

Weakly a la Tarski

Weakly a la Tarski, the type of all entire relations is given by the following rules:

Formation rules:

ΓAtypeΓBtypeΓEntRel(A,B)type\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{EntRel}(A, B) \; \mathrm{type}}

Introduction rules:

ΓAtypeΓBtypeΓ,x:A,y:BR(x,y)ΓtoElem A,B,R:isEntire(x:A.y:B.R(x,y))EntRel(A,B)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y)}{\Gamma \vdash \mathrm{toElem}_{A, B, R}:\mathrm{isEntire}(x:A.y:B.R(x, y)) \to \mathrm{EntRel}(A, B)}

Elimination rules:

ΓAtypeΓBtypeΓR:EntRel(A,B)Γ,x:A,y:BEl(R,x,y)type\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma, x:A, y:B \vdash \mathrm{El}(R, x, y) \; \mathrm{type}}
ΓAtypeΓBtypeΓR:EntRel(A,B)Γentrelwitn(R):isEntire(x:A.y:B.El(R,x,y))\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{entrelwitn}(R):\mathrm{isEntire}(x:A.y:B.\mathrm{El}(R, x, y))}

Computation rules:

ΓAtypeΓBtypeΓ,x:A,y:BR(x,y)Γp:isEntire(x:A.y:B.R(x,y))Γ,x:A,y:Bβ EntRel(A,B) El,R(p,x,y):El(toElem A,B,R(p),x,y)R(x,y)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma, x:A, y:B \vdash \beta_{\mathrm{EntRel}(A, B)}^{\mathrm{El}, R}(p, x, y):\mathrm{El}(\mathrm{toElem}_{A, B, R}(p), x, y) \simeq R(x, y)}
  • Judgmental computation rules
ΓAtypeΓBtypeΓ,x:A,y:BR(x,y)Γp:isEntire(x:A.y:B.R(x,y))Γ(congform isEntire(β EntRel(A,B) El,R(p)))(entrelwitn(toElem A,B,R(p)))p:isEntire(x:A.y:B.R(x,y))\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma \vdash (\mathrm{congform}_{\mathrm{isEntire}}(\beta_{\mathrm{EntRel}(A, B)}^{\mathrm{El}, R}(p)))(\mathrm{entrelwitn}(\mathrm{toElem}_{A, B, R}(p))) \equiv p:\mathrm{isEntire}(x:A.y:B.R(x, y))}
  • Typal computation rules
ΓAtypeΓBtypeΓ,x:A,y:BR(x,y)Γp:isEntire(x:A.y:B.R(x,y))Γβ EntRel(A,B) entrelwitn,R:(congform isEntire(β EntRel(A,B) El,R(p)))(entrelwitn(toElem A,B,R(p)))= isEntire(x:A.y:B.R(x,y))p\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma \vdash \beta_{\mathrm{EntRel}(A, B)}^{\mathrm{entrelwitn}, R}:(\mathrm{congform}_{\mathrm{isEntire}}(\beta_{\mathrm{EntRel}(A, B)}^{\mathrm{El}, R}(p)))(\mathrm{entrelwitn}(\mathrm{toElem}_{A, B, R}(p))) =_{\mathrm{isEntire}(x:A.y:B.R(x, y))} p}

where the equivalence

congform isEntire(β EntRel(A,B) El,R(p)):isEntire(x:A.y:B.El(R,x,y))isEntire(x:A.y:B.R(x,y))\mathrm{congform}_{\mathrm{isEntire}}(\beta_{\mathrm{EntRel}(A, B)}^{\mathrm{El}, R}(p)):\mathrm{isEntire}(x:A.y:B.\mathrm{El}(R, x, y)) \simeq \mathrm{isEntire}(x:A.y:B.R(x, y))

is provided from the typal computation rules for isEntire(x:A.y:B.R(x,y))\mathrm{isEntire}(x:A.y:B.R(x, y)).

Uniqueness rules:

  • Judgmental uniqueness rules:
ΓAtypeΓBtypeΓR:EntRel(A,B)ΓtoElem A,B,El(R)(entrelwitn(R))R:EntRel(A,B)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{toElem}_{A, B, \mathrm{El}(R)}(\mathrm{entrelwitn}(R)) \equiv R:\mathrm{EntRel}(A, B)}
  • Typal uniqueness rules:
ΓAtypeΓBtypeΓR:EntRel(A,B)Γη EntRel(A,B)(R):toElem A,B,El(R)(entrelwitn(R))= EntRel(A,B)R\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \eta_{\mathrm{EntRel}(A, B)}(R):\mathrm{toElem}_{A, B, \mathrm{El}(R)}(\mathrm{entrelwitn}(R)) =_{\mathrm{EntRel}(A, B)} R}

Extensionality rule:

ΓAtypeΓBtypeΓR:EntRel(A,B)ΓS:EntRel(A,B)Γentrelext A,B(R,S):(R= EntRel(A,B)S) x:A y:BEl(R,x,y)El(S,x,y)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B) \quad \Gamma \vdash S:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{entrelext}_{A, B}(R, S):(R =_{\mathrm{EntRel}(A, B)} S) \simeq \prod_{x:A} \prod_{y:B} \mathrm{El}(R, x, y) \simeq \mathrm{El}(S, x, y)}

Last revised on November 24, 2023 at 21:31:13. See the history of this page for a list of all contributions to it.