Nuprl Lemma : csm-fibrant-type-id
∀[G:j⊢]. ∀[FT:FibrantType(G)]. ∀[tau:G j⟶ G].
csm-fibrant-type(G;G;tau;FT) = FT ∈ FibrantType(G) supposing tau = 1(G) ∈ G j⟶ G
Proof
Definitions occuring in Statement :
csm-fibrant-type: csm-fibrant-type(G;H;s;FT),
fibrant-type: FibrantType(X),
csm-id: 1(X),
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
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,
fibrant-type: FibrantType(X),
csm-fibrant-type: csm-fibrant-type(G;H;s;FT),
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
csm-ap-id-type,
csm-composition-id,
csm-ap-type_wf,
csm-id_wf,
subtype_rel-equal,
composition-op_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
subtype_rel_self,
iff_weakening_equal,
fibrant-type_wf,
csm-fibrant-type_wf,
cube_set_map_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
thin,
sqequalHypSubstitution,
productElimination,
sqequalRule,
dependent_pairEquality_alt,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
equalityIstype,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[FT:FibrantType(G)]. \mforall{}[tau:G j{}\mrightarrow{} G].
csm-fibrant-type(G;G;tau;FT) = FT supposing tau = 1(G)
Date html generated:
2020_05_20-PM-05_20_38
Last ObjectModification:
2020_04_12-AM-08_43_18
Theory : cubical!type!theory
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