Nuprl Lemma : contractible-comp_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[cA:X ⊢ CompOp(A)]. (contractible-comp(X;A;cA) ∈ X ⊢ CompOp(Contractible(A)))
Proof
Definitions occuring in Statement :
contractible-comp: contractible-comp(X;A;cA),
composition-op: Gamma ⊢ CompOp(A),
contractible-type: Contractible(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,
contractible-comp: contractible-comp(X;A;cA),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B
Lemmas referenced :
comp-fun-to-comp-op_wf,
contractible-type_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
contractible_comp_wf,
comp-op-to-comp-fun_wf2,
composition-op_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
applyEquality,
hypothesis,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[cA:X \mvdash{} CompOp(A)].
(contractible-comp(X;A;cA) \mmember{} X \mvdash{} CompOp(Contractible(A)))
Date html generated:
2020_05_20-PM-05_14_17
Last ObjectModification:
2020_04_10-AM-11_45_59
Theory : cubical!type!theory
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