Nuprl Lemma : cu-cube-restriction-comp
∀[I:Cname List]. ∀[alpha:c𝕌(I)]. ∀[L,J,K:Cname List]. ∀[f:name-morph(L;J)]. ∀[g:name-morph(J;K)]. ∀[a:name-morph(I;L)].
∀[T:cu-cube-family(alpha;L;a)].
(cu-cube-restriction(alpha;J;K;g;(a o f);cu-cube-restriction(alpha;L;J;f;a;T))
= cu-cube-restriction(alpha;L;K;(f o g);a;T)
∈ cu-cube-family(alpha;K;(a o (f o g))))
Proof
Definitions occuring in Statement :
cu-cube-restriction: cu-cube-restriction(alpha;L;J;f;a;T),
cu-cube-family: cu-cube-family(alpha;L;f),
cubical-universe: c𝕌,
I-cube: X(I),
name-comp: (f o g),
name-morph: name-morph(I;J),
coordinate_name: Cname,
list: T List,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
pi1: fst(t),
cu-cube-family: cu-cube-family(alpha;L;f),
cu-cube-restriction: cu-cube-restriction(alpha;L;J;f;a;T),
pi2: snd(t),
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cubical-universe-I-cube,
equal_wf,
squash_wf,
true_wf,
list_wf,
coordinate_name_wf,
name-comp_wf,
iff_weakening_equal,
cu-cube-family_wf,
name-morph_wf,
I-cube_wf,
cubical-universe_wf
Rules used in proof :
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productElimination,
sqequalRule,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
functionExtensionality,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
instantiate,
isect_memberFormation,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[I:Cname List]. \mforall{}[alpha:c\mBbbU{}(I)]. \mforall{}[L,J,K:Cname List]. \mforall{}[f:name-morph(L;J)]. \mforall{}[g:name-morph(J;K)].
\mforall{}[a:name-morph(I;L)]. \mforall{}[T:cu-cube-family(alpha;L;a)].
(cu-cube-restriction(alpha;J;K;g;(a o f);cu-cube-restriction(alpha;L;J;f;a;T))
= cu-cube-restriction(alpha;L;K;(f o g);a;T))
Date html generated:
2017_10_05-PM-04_13_33
Last ObjectModification:
2017_07_28-AM-11_30_12
Theory : cubical!sets
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