Nuprl Lemma : cubical-beta
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[b:{X.A ⊢ _:B}]. ∀[u:{X ⊢ _:A}].
  (app((λb); u) = (b)[u] ∈ {X ⊢ _:(B)[u]})
Proof
Definitions occuring in Statement : 
cubical-app: app(w; u)
, 
cubical-lambda: (λb)
, 
csm-id-adjoin: [u]
, 
cube-context-adjoin: X.A
, 
csm-ap-term: (t)s
, 
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
, 
uimplies: b supposing a
, 
cubical-lambda: (λb)
, 
cubical-app: app(w; u)
, 
csm-ap-term: (t)s
, 
csm-ap: (s)x
, 
cubical-term: {X ⊢ _:AF}
, 
cubical-type-at: A(a)
, 
csm-id-adjoin: [u]
, 
csm-id: 1(X)
, 
csm-adjoin: (s;u)
, 
type-cat: TypeCat
, 
identity-trans: identity-trans(C;D;F)
, 
all: ∀x:A. B[x]
, 
top: Top
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
cc-adjoin-cube: (v;u)
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
squash: ↓T
, 
subtype_rel: A ⊆r B
, 
true: True
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
rev_implies: P 
⇐ Q
, 
cubical-type: {X ⊢ _}
, 
csm-ap-type: (AF)s
, 
pi1: fst(t)
Lemmas referenced : 
cubical-term-equal, 
csm-ap-type_wf, 
cube-context-adjoin_wf, 
csm-id-adjoin_wf, 
csm-ap-term_wf, 
cubical-term_wf, 
cubical-type_wf, 
cubical-set_wf, 
ap_mk_nat_trans_lemma, 
cat_id_tuple_lemma, 
list_wf, 
coordinate_name_wf, 
cubical-type-at_wf, 
equal_wf, 
I-cube_wf, 
squash_wf, 
true_wf, 
cc-adjoin-cube_wf, 
cube-set-restriction-id, 
subtype_rel-equal, 
cube-set-restriction_wf, 
id-morph_wf, 
iff_weakening_equal, 
subtype_rel_self, 
subtype_rel_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
equalitySymmetry, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
equalityTransitivity, 
independent_isectElimination, 
sqequalRule, 
isect_memberEquality, 
axiomEquality, 
because_Cache, 
functionExtensionality, 
rename, 
setElimination, 
dependent_functionElimination, 
voidElimination, 
voidEquality, 
applyEquality, 
lambdaFormation, 
independent_functionElimination, 
lambdaEquality, 
imageElimination, 
universeEquality, 
instantiate, 
equalityUniverse, 
levelHypothesis, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
productElimination, 
hyp_replacement, 
applyLambdaEquality
Latex:
\mforall{}[X:CubicalSet].  \mforall{}[A:\{X  \mvdash{}  \_\}].  \mforall{}[B:\{X.A  \mvdash{}  \_\}].  \mforall{}[b:\{X.A  \mvdash{}  \_:B\}].  \mforall{}[u:\{X  \mvdash{}  \_:A\}].
    (app((\mlambda{}b);  u)  =  (b)[u])
Date html generated:
2017_10_05-AM-10_17_10
Last ObjectModification:
2017_07_28-AM-11_20_08
Theory : cubical!sets
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