Nuprl Lemma : csm-ap-cubical-app
∀[X,Delta:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}]. ∀[u:{X ⊢ _:A}]. ∀[s:Delta ⟶ X].
((app(w; u))s = app((w)s; (u)s) ∈ {Delta ⊢ _:((B)[u])s})
Proof
Definitions occuring in Statement :
cubical-app: app(w; u),
cubical-pi: ΠA B,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:AF},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
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,
uimplies: b supposing a,
cubical-type: {X ⊢ _},
cubical-term: {X ⊢ _:AF},
cube-set-map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
csm-ap-term: (t)s,
cubical-app: app(w; u),
csm-id-adjoin: [u],
csm-ap-type: (AF)s,
pi1: fst(t),
csm-ap: (s)x,
csm-adjoin: (s;u),
cubical-set: CubicalSet,
functor-arrow: arrow(F),
functor-ob: ob(F),
type-cat: TypeCat,
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
name-cat: NameCat,
cat-ob: cat-ob(C),
pi2: snd(t),
I-cube: X(I),
cubical-pi: ΠA B,
csm-id: 1(X),
all: ∀x:A. B[x],
top: Top,
cat-id: cat-id(C),
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
and: P ∧ Q,
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
cubical-type-at: A(a),
squash: ↓T,
label: ...$L... t,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cc-adjoin-cube: (v;u)
Lemmas referenced :
cubical-term-equal,
csm-ap-type_wf,
cube-context-adjoin_wf,
csm-id-adjoin_wf,
csm-ap-term_wf,
cubical-app_wf,
cube-set-map_wf,
cubical-term_wf,
cubical-pi_wf,
cubical-type_wf,
cubical-set_wf,
ob_pair_lemma,
ident_trans_ap_lemma,
list_wf,
coordinate_name_wf,
I-cube_wf,
equal_wf,
cubical-pi-family_wf,
id-morph_wf,
subtype_rel-equal,
cube-set-restriction_wf,
squash_wf,
true_wf,
cube-set-restriction-id,
subtype_rel_self,
iff_weakening_equal,
cc-adjoin-cube_wf,
cubical-type-at_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
setElimination,
rename,
productElimination,
dependent_functionElimination,
voidElimination,
voidEquality,
lambdaEquality,
applyEquality,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
functionExtensionality,
instantiate,
imageElimination,
universeEquality,
hyp_replacement,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[X,Delta:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:\mPi{}A B\}]. \mforall{}[u:\{X \mvdash{} \_:A\}].
\mforall{}[s:Delta {}\mrightarrow{} X].
((app(w; u))s = app((w)s; (u)s))
Date html generated:
2018_05_23-PM-06_31_54
Last ObjectModification:
2018_05_20-PM-04_20_44
Theory : cubical!sets
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