Nuprl Lemma : cubical-pair-eta
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:Σ A B}]. (cubical-pair(w.1;w.2) = w ∈ {X ⊢ _:Σ A B})
Proof
Definitions occuring in Statement :
cubical-pair: cubical-pair(u;v),
cubical-snd: p.2,
cubical-fst: p.1,
cubical-sigma: Σ A B,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:AF},
cubical-type: {X ⊢ _},
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
cubical-term: {X ⊢ _:AF},
cubical-snd: p.2,
cubical-fst: p.1,
cubical-pair: cubical-pair(u;v),
cubical-sigma: Σ A B,
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
pi2: snd(t),
uimplies: b supposing a,
and: P ∧ Q
Lemmas referenced :
cubical-term-equal,
cubical-sigma_wf,
cubical-pair_wf,
cubical-fst_wf,
cubical-snd_wf,
subtype_rel-equal,
cubical-type-at_wf,
cube-context-adjoin_wf,
cc-adjoin-cube_wf,
I-cube_wf,
list_wf,
coordinate_name_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
equalitySymmetry,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
because_Cache,
sqequalRule,
functionExtensionality_alt,
independent_pairEquality,
inhabitedIsType,
lambdaFormation_alt,
productElimination,
equalityIstype,
equalityTransitivity,
dependent_functionElimination,
independent_functionElimination,
dependent_pairEquality_alt,
independent_isectElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
applyLambdaEquality,
universeIsType
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:\mSigma{} A B\}]. (cubical-pair(w.1;w.2) = w)
Date html generated:
2020_05_21-AM-10_51_23
Last ObjectModification:
2020_01_01-PM-02_52_02
Theory : cubical!sets
Home
Index