Nuprl Lemma : csm-Kan-id
∀[Gamma:CubicalSet]. ∀[AK:{Gamma ⊢ _(Kan)}]. ((AK)1(Gamma) = AK ∈ {Gamma ⊢ _(Kan)})
Proof
Definitions occuring in Statement :
csm-Kan-cubical-type: (AK)s,
Kan-cubical-type: {X ⊢ _(Kan)},
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,
Kan-cubical-type: {X ⊢ _(Kan)},
cubical-set: CubicalSet,
csm-Kan-cubical-type: (AK)s,
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
nameset: nameset(L),
csm-ap: (s)x,
csm-id: 1(X),
identity-trans: identity-trans(C;D;F),
mk-nat-trans: x |→ T[x],
cat-id: cat-id(C),
pi1: fst(t),
pi2: snd(t),
type-cat: TypeCat
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
cubical-type_wf,
csm-ap-id-type,
iff_weakening_equal,
list_wf,
coordinate_name_wf,
I-cube_wf,
nameset_wf,
int_seg_wf,
A-open-box_wf,
subtype_rel_list,
cubical-type-at_wf,
Kan-A-filler_wf,
uniform-Kan-A-filler_wf,
Kan-cubical-type_wf,
cubical-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
equalitySymmetry,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
productElimination,
hypothesisEquality,
sqequalRule,
dependent_pairEquality,
applyEquality,
instantiate,
lambdaEquality,
imageElimination,
extract_by_obid,
isectElimination,
equalityTransitivity,
hypothesis,
universeEquality,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
functionEquality,
dependent_functionElimination,
productEquality,
functionExtensionality,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[Gamma:CubicalSet]. \mforall{}[AK:\{Gamma \mvdash{} \_(Kan)\}]. ((AK)1(Gamma) = AK)
Date html generated:
2017_10_05-AM-10_23_54
Last ObjectModification:
2017_07_28-AM-11_22_03
Theory : cubical!sets
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