Nuprl Lemma : cubical-fst-pair
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[v:{X ⊢ _:(B)[u]}].
(cubical-pair(u;v).1 = u ∈ {X ⊢ _:A})
Proof
Definitions occuring in Statement :
cubical-pair: cubical-pair(u;v),
cubical-fst: p.1,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:AF},
csm-ap-type: (AF)s,
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},
uimplies: b supposing a,
cubical-pair: cubical-pair(u;v),
cubical-fst: p.1,
pi1: fst(t),
cubical-term-at: u(a),
cubical-type-at: A(a)
Lemmas referenced :
cubical-term-equal,
cubical-fst_wf,
cubical-pair_wf,
cubical-term_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
csm-id-adjoin_wf,
cubical-type_wf,
cubical-set_wf,
cubical-term-at_wf,
I-cube_wf,
list_wf,
coordinate_name_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,
independent_isectElimination,
universeIsType,
functionExtensionality
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[u:\{X \mvdash{} \_:A\}]. \mforall{}[v:\{X \mvdash{} \_:(B)[u]\}].
(cubical-pair(u;v).1 = u)
Date html generated:
2020_05_21-AM-10_51_19
Last ObjectModification:
2020_01_05-AM-00_22_20
Theory : cubical!sets
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