Nuprl Lemma : csm-fibrant-comp
∀[X,Y,Z:j⊢]. ∀[g:Z j⟶ Y]. ∀[f:Y j⟶ X]. ∀[FT:FibrantType(X)].
(csm-fibrant-type(X;Z;f o g;FT) = csm-fibrant-type(Y;Z;g;csm-fibrant-type(X;Y;f;FT)) ∈ FibrantType(Z))
Proof
Definitions occuring in Statement :
csm-fibrant-type: csm-fibrant-type(G;H;s;FT),
fibrant-type: FibrantType(X),
csm-comp: G o F,
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,
fibrant-type: FibrantType(X),
csm-fibrant-type: csm-fibrant-type(G;H;s;FT),
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
csm-comp: G o F,
csm-ap: (s)x,
compose: f o g,
subtype_rel: A ⊆r B
Lemmas referenced :
csm-ap-type_wf,
csm-composition-comp,
composition-op_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
fibrant-type_wf,
cube_set_map_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
rename,
sqequalRule,
dependent_pairEquality_alt,
setElimination,
hypothesis,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalitySymmetry,
universeIsType,
instantiate,
applyEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
because_Cache
Latex:
\mforall{}[X,Y,Z:j\mvdash{}]. \mforall{}[g:Z j{}\mrightarrow{} Y]. \mforall{}[f:Y j{}\mrightarrow{} X]. \mforall{}[FT:FibrantType(X)].
(csm-fibrant-type(X;Z;f o g;FT) = csm-fibrant-type(Y;Z;g;csm-fibrant-type(X;Y;f;FT)))
Date html generated:
2020_05_20-PM-05_20_52
Last ObjectModification:
2020_04_12-AM-08_42_59
Theory : cubical!type!theory
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