Nuprl Lemma : singleton-contraction_wf2
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[a:{X ⊢ _:T}]. ∀[s:{X ⊢ _:Singleton(a)}].
(singleton-contraction(X;s.2) ∈ {X ⊢ _:(Path_Singleton(a) singleton-center(X;a) s)})
Proof
Definitions occuring in Statement :
singleton-center: singleton-center(X;a),
singleton-type: Singleton(a),
singleton-contraction: singleton-contraction(X;pth),
path-type: (Path_A a b),
cubical-snd: p.2,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
singleton-type: Singleton(a),
singleton-center: singleton-center(X;a),
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
cubical-sigma: Σ A B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True
Lemmas referenced :
cubical-term_wf,
singleton-type_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
singleton-contraction_wf,
cubical-fst_wf,
path-type_wf,
cube-context-adjoin_wf,
csm-ap-type_wf,
cc-fst_wf,
csm-ap-term_wf,
cc-snd_wf,
cubical-snd_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-id-adjoin_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
singleton-center_wf,
cubical-pair-eta,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
thin,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
natural_numberEquality,
hyp_replacement
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T:\{X \mvdash{} \_\}]. \mforall{}[a:\{X \mvdash{} \_:T\}]. \mforall{}[s:\{X \mvdash{} \_:Singleton(a)\}].
(singleton-contraction(X;s.2) \mmember{} \{X \mvdash{} \_:(Path\_Singleton(a) singleton-center(X;a) s)\})
Date html generated:
2020_05_20-PM-03_29_49
Last ObjectModification:
2020_04_06-PM-06_51_51
Theory : cubical!type!theory
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