Nuprl Lemma : cubical-app_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}]. ∀[u:{X ⊢ _:A}]. (app(w; u) ∈ {X ⊢ _:(B)[u]})
Proof
Definitions occuring in Statement :
cubical-app: app(w; u),
cubical-pi: ΠA B,
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],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-term: {X ⊢ _:AF},
all: ∀x:A. B[x],
implies: P ⇒ Q,
cubical-type: {X ⊢ _},
pi1: fst(t),
cubical-app: app(w; u),
cubical-type-at: A(a),
csm-ap-type: (AF)s,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cubical-pi: ΠA B,
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
pi2: snd(t),
cubical-type-ap-morph: (u a f),
top: Top,
cube-set-restriction: f(s),
type-cat: TypeCat,
cat-id: cat-id(C),
mk-nat-trans: x |→ T[x],
identity-trans: identity-trans(C;D;F),
csm-id: 1(X),
cc-adjoin-cube: (v;u),
csm-adjoin: (s;u),
csm-ap: (s)x,
csm-id-adjoin: [u],
so_apply: x[s],
so_lambda: λ2x.t[x],
cand: A c∧ B
Lemmas referenced :
csm-ap-type_wf,
cube-context-adjoin_wf,
csm-id-adjoin_wf,
list_wf,
coordinate_name_wf,
name-morph_wf,
I-cube_wf,
cube-set-restriction_wf,
cubical-term_wf,
cubical-pi_wf,
cubical-type_wf,
cubical-set_wf,
id-morph_wf,
csm-ap_wf,
subtype_rel-equal,
cubical-type-at_wf,
cube-set-restriction-id,
cc-adjoin-cube_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
csm-id-adjoin-ap,
subtype_rel_weakening,
ext-eq_weakening,
subtype_rel_wf,
cubical-pi-family_wf,
cc-adjoin-cube-restriction,
name-comp_wf,
subtype_rel_dep_function,
name-comp-id-left,
csm-ap-restriction,
ap_mk_nat_trans_lemma,
name-comp-id-right
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
setElimination,
rename,
productElimination,
sqequalRule,
functionIsType,
universeIsType,
because_Cache,
equalityIstype,
applyEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
lambdaEquality_alt,
natural_numberEquality,
independent_isectElimination,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
lambdaFormation,
promote_hyp,
applyLambdaEquality,
functionEquality,
lambdaEquality,
voidEquality,
voidElimination,
isect_memberEquality,
comment,
productEquality,
independent_pairFormation,
dependent_set_memberEquality,
functionExtensionality,
hyp_replacement,
setEquality
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:\mPi{}A B\}]. \mforall{}[u:\{X \mvdash{} \_:A\}].
(app(w; u) \mmember{} \{X \mvdash{} \_:(B)[u]\})
Date html generated:
2020_05_21-AM-10_51_11
Last ObjectModification:
2020_01_01-PM-02_49_36
Theory : cubical!sets
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