Nuprl Lemma : csm-ap-id-type
∀[Gamma:CubicalSet]. ∀[A:{Gamma ⊢ _}]. ((A)1(Gamma) = A ∈ {Gamma ⊢ _})
Proof
Definitions occuring in Statement :
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
csm-id: 1(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,
cubical-type: {X ⊢ _},
and: P ∧ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_apply: x[s],
all: ∀x:A. B[x],
cubical-set: CubicalSet,
csm-id: 1(X),
csm-ap-type: (AF)s,
type-cat: TypeCat,
identity-trans: identity-trans(C;D;F),
csm-ap: (s)x,
top: Top,
I-cube: X(I),
functor-ob: ob(F),
pi1: fst(t)
Lemmas referenced :
all_wf,
list_wf,
coordinate_name_wf,
I-cube_wf,
equal_wf,
id-morph_wf,
subtype_rel-equal,
cube-set-restriction_wf,
squash_wf,
true_wf,
cube-set-restriction-id,
iff_weakening_equal,
name-morph_wf,
name-comp_wf,
cube-set-restriction-comp,
cubical-type_wf,
cubical-set_wf,
ap_mk_nat_trans_lemma,
cat_id_tuple_lemma,
ob_pair_lemma,
eta_conv,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
equalitySymmetry,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
hypothesis,
productElimination,
sqequalRule,
productEquality,
extract_by_obid,
isectElimination,
lambdaEquality,
hypothesisEquality,
applyEquality,
functionExtensionality,
because_Cache,
independent_isectElimination,
instantiate,
imageElimination,
equalityTransitivity,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
voidElimination,
voidEquality,
dependent_pairEquality,
functionEquality,
cumulativity
Latex:
\mforall{}[Gamma:CubicalSet]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. ((A)1(Gamma) = A)
Date html generated:
2017_10_05-AM-10_12_52
Last ObjectModification:
2017_07_28-AM-11_18_33
Theory : cubical!sets
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