Nuprl Lemma : csm-Kan-unit-cube-comp
∀I,J,K:Cname List. ∀f:name-morph(I;J). ∀g:name-morph(J;K). ∀x:{unit-cube(I) ⊢ _(Kan)}.
((x)unit-cube-map((f o g)) = ((x)unit-cube-map(f))unit-cube-map(g) ∈ {unit-cube(K) ⊢ _(Kan)})
Proof
Definitions occuring in Statement :
csm-Kan-cubical-type: (AK)s,
Kan-cubical-type: {X ⊢ _(Kan)},
unit-cube-map: unit-cube-map(f),
unit-cube: unit-cube(I),
name-comp: (f o g),
name-morph: name-morph(I;J),
coordinate_name: Cname,
list: T List,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ
Lemmas referenced :
equal_wf,
cube-set-map_wf,
unit-cube_wf,
unit-cube-map-comp,
csm-comp_wf,
unit-cube-map_wf,
iff_weakening_equal,
Kan-cubical-type_wf,
csm-Kan-comp,
csm-Kan-cubical-type_wf,
name-morph_wf,
list_wf,
coordinate_name_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
thin,
instantiate,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}I,J,K:Cname List. \mforall{}f:name-morph(I;J). \mforall{}g:name-morph(J;K). \mforall{}x:\{unit-cube(I) \mvdash{} \_(Kan)\}.
((x)unit-cube-map((f o g)) = ((x)unit-cube-map(f))unit-cube-map(g))
Date html generated:
2017_10_05-AM-10_24_09
Last ObjectModification:
2017_07_28-AM-11_22_16
Theory : cubical!sets
Home
Index