Nuprl Lemma : csm-singleton-center
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ∀[H:j⊢]. ∀[s:H j⟶ X].
((singleton-center(X;a))s = singleton-center(H;(a)s) ∈ {H ⊢ _:Singleton((a)s)})
Proof
Definitions occuring in Statement :
singleton-center: singleton-center(X;a),
singleton-type: Singleton(a),
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
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,
singleton-center: singleton-center(X;a),
singleton-type: Singleton(a),
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
all: ∀x:A. B[x],
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
csm-id: 1(X),
csm-ap: (s)x
Lemmas referenced :
csm-cubical-pair,
cubical-pair_wf,
squash_wf,
true_wf,
cubical-term_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
csm-id-adjoin_wf,
cubical-type_wf,
path-type_wf,
cc-fst_wf,
csm-ap-term_wf,
cc-snd_wf,
cube_set_map_wf,
cubical_set_wf,
csm-cubical-refl,
equal_wf,
istype-universe,
csm-path-type,
subtype_rel_self,
iff_weakening_equal,
csm_id_adjoin_fst_type_lemma,
csm_id_adjoin_fst_term_lemma,
cc_snd_csm_id_adjoin_lemma,
csm-id_wf,
csm-ap-id-term,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-id-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
applyEquality,
lambdaEquality_alt,
imageElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
instantiate,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
hyp_replacement,
universeEquality,
dependent_functionElimination,
independent_isectElimination,
productElimination,
independent_functionElimination,
setElimination,
rename
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a:\{X \mvdash{} \_:A\}]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} X].
((singleton-center(X;a))s = singleton-center(H;(a)s))
Date html generated:
2020_05_20-PM-03_29_37
Last ObjectModification:
2020_04_06-PM-06_51_35
Theory : cubical!type!theory
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