Nuprl Lemma : cubical-term-at-morph
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[I:Cname List]. ∀[a:X(I)]. ∀[J:Cname List]. ∀[f:name-morph(I;J)].
((u(a) a f) = u(f(a)) ∈ A(f(a)))
Proof
Definitions occuring in Statement :
cubical-term-at: u(a),
cubical-term: {X ⊢ _:AF},
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
name-morph: name-morph(I;J),
coordinate_name: Cname,
list: T List,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-type: {X ⊢ _},
cubical-term: {X ⊢ _:AF},
pi1: fst(t),
cubical-term-at: u(a),
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
pi2: snd(t),
all: ∀x:A. B[x]
Lemmas referenced :
name-morph_wf,
list_wf,
coordinate_name_wf,
I-cube_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
lemma_by_obid,
isectElimination,
isect_memberEquality,
axiomEquality,
because_Cache
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[u:\{X \mvdash{} \_:A\}]. \mforall{}[I:Cname List]. \mforall{}[a:X(I)]. \mforall{}[J:Cname List].
\mforall{}[f:name-morph(I;J)].
((u(a) a f) = u(f(a)))
Date html generated:
2016_06_16-PM-05_40_10
Last ObjectModification:
2015_12_28-PM-04_35_37
Theory : cubical!sets
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