Nuprl Lemma : cubical-type-ap-morph-id
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[I:Cname List]. ∀[f:name-morph(I;I)]. ∀[a:X(I)]. ∀[u:A(a)].
(u a f) = u ∈ A(a) supposing f = 1 ∈ name-morph(I;I)
Proof
Definitions occuring in Statement :
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
I-cube: X(I),
cubical-set: CubicalSet,
id-morph: 1,
name-morph: name-morph(I;J),
coordinate_name: Cname,
list: T List,
uimplies: b supposing a,
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,
prop: ℙ,
cubical-type: {X ⊢ _},
subtype_rel: A ⊆r B,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True,
cubical-type-at: A(a),
pi1: fst(t),
cubical-type-ap-morph: (u a f),
pi2: snd(t),
all: ∀x:A. B[x]
Lemmas referenced :
equal_wf,
name-morph_wf,
id-morph_wf,
cubical-type-at_wf,
cubical-type-ap-morph_wf,
subtype_rel-equal,
cube-set-restriction_wf,
squash_wf,
true_wf,
I-cube_wf,
cube-set-restriction-when-id,
subtype_rel_self,
iff_weakening_equal,
list_wf,
coordinate_name_wf,
cubical-type_wf,
cubical-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
hypothesis,
because_Cache,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
productElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
applyEquality,
independent_isectElimination,
lambdaEquality,
imageElimination,
equalityTransitivity,
universeEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
instantiate,
independent_functionElimination,
natural_numberEquality,
isect_memberEquality,
axiomEquality,
dependent_functionElimination
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[I:Cname List]. \mforall{}[f:name-morph(I;I)]. \mforall{}[a:X(I)]. \mforall{}[u:A(a)].
(u a f) = u supposing f = 1
Date html generated:
2018_05_23-PM-06_28_40
Last ObjectModification:
2018_05_20-PM-04_08_45
Theory : cubical!sets
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