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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