Nuprl Lemma : csm-Kan-comp
∀[Gamma,Delta,Z:CubicalSet]. ∀[s1:Z ⟶ Delta]. ∀[s2:Delta ⟶ Gamma]. ∀[AK:{Gamma ⊢ _(Kan)}].
((AK)s2 o s1 = ((AK)s2)s1 ∈ {Z ⊢ _(Kan)})
Proof
Definitions occuring in Statement :
csm-Kan-cubical-type: (AK)s,
Kan-cubical-type: {X ⊢ _(Kan)},
csm-comp: G o F,
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
Kan-cubical-type: {X ⊢ _(Kan)},
uimplies: b supposing a,
csm-Kan-cubical-type: (AK)s,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
nameset: nameset(L),
so_apply: x[s]
Lemmas referenced :
Kan-cubical-type-equal,
csm-Kan-cubical-type_wf,
csm-comp_wf,
Kan-cubical-type_wf,
cube-set-map_wf,
equal_wf,
squash_wf,
true_wf,
cubical-type_wf,
csm-ap-comp-type,
csm-ap-type_wf,
iff_weakening_equal,
list_wf,
coordinate_name_wf,
csm-ap-csm-comp,
csm-ap_wf,
subtype_rel_dep_function,
nameset_wf,
int_seg_wf,
A-open-box_wf,
subtype_rel_list,
cubical-type-at_wf,
csm-A-open-box,
subtype_rel-equal,
csm-type-at,
I-cube_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
independent_isectElimination,
isect_memberEquality,
axiomEquality,
because_Cache,
productElimination,
dependent_pairEquality,
instantiate,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
functionExtensionality,
dependent_functionElimination,
voidElimination,
voidEquality,
functionEquality,
lambdaFormation
Latex:
\mforall{}[Gamma,Delta,Z:CubicalSet]. \mforall{}[s1:Z {}\mrightarrow{} Delta]. \mforall{}[s2:Delta {}\mrightarrow{} Gamma]. \mforall{}[AK:\{Gamma \mvdash{} \_(Kan)\}].
((AK)s2 o s1 = ((AK)s2)s1)
Date html generated:
2017_10_05-AM-10_23_59
Last ObjectModification:
2017_07_28-AM-11_22_08
Theory : cubical!sets
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