Nuprl Lemma : cubical-type-ap-morph-id
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[I:fset(ℕ)]. ∀[f:I ⟶ I]. ∀[a:X(I)]. ∀[u:A(a)]. (u a f) = u ∈ A(a) supposing f = 1 ∈ I ⟶ I
Proof
Definitions occuring in Statement :
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
I_cube: A(I),
cubical_set: CubicalSet,
nh-id: 1,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
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,
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: {X ⊢ _},
cubical-type-ap-morph: (u a f),
all: ∀x:A. B[x],
pi2: snd(t)
Lemmas referenced :
equal_wf,
cubical-type-at_wf,
cubical-type-ap-morph_wf,
subtype_rel-equal,
cube-set-restriction_wf,
squash_wf,
true_wf,
istype-universe,
I_cube_wf,
cube-set-restriction-when-id,
subtype_rel_self,
iff_weakening_equal,
nh-id_wf,
istype-cubical-type-at,
names-hom_wf,
fset_wf,
nat_wf,
cubical-type_wf,
cubical_set_wf,
cubical_type_at_pair_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
hypothesis,
because_Cache,
equalityIstype,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
thin,
hyp_replacement,
applyLambdaEquality,
setElimination,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
instantiate,
universeIsType,
universeEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
natural_numberEquality,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
dependent_functionElimination,
Error :memTop
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[f:I {}\mrightarrow{} I]. \mforall{}[a:X(I)]. \mforall{}[u:A(a)]. (u a f) = u supposing f = 1
Date html generated:
2020_05_20-PM-01_48_42
Last ObjectModification:
2020_04_17-PM-04_22_07
Theory : cubical!type!theory
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