Nuprl Lemma : contractible-to-prop_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[cA:X +⊢ Compositon(A)]. ∀[c:{X ⊢ _:Contractible(A)}].
(contractible-to-prop(X;A;cA;c) ∈ {X ⊢ _:isProp(A)})
Proof
Definitions occuring in Statement :
contractible-to-prop: contractible-to-prop(X;A;cA;c),
composition-structure: Gamma ⊢ Compositon(A),
is-prop: isProp(A),
contractible-type: Contractible(A),
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,
squash: ↓T,
prop: ℙ,
true: True,
is-prop: isProp(A),
contractible-to-prop: contractible-to-prop(X;A;cA;c),
guard: {T}
Lemmas referenced :
csm-ap-term_wf,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
csm-ap-type_wf,
cc-fst_wf,
contractible-type_wf,
cubical-term_wf,
squash_wf,
true_wf,
csm-contractible-type,
contr-path_wf,
contr-center_wf,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
cubical-lambda_wf,
cubical-pi_wf,
path-type_wf,
cc-snd_wf,
comp-path_wf,
csm-comp-structure_wf,
rev-path_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
universeIsType,
applyLambdaEquality,
inhabitedIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[cA:X +\mvdash{} Compositon(A)]. \mforall{}[c:\{X \mvdash{} \_:Contractible(A)\}].
(contractible-to-prop(X;A;cA;c) \mmember{} \{X \mvdash{} \_:isProp(A)\})
Date html generated:
2020_05_20-PM-04_58_23
Last ObjectModification:
2020_04_13-PM-02_15_04
Theory : cubical!type!theory
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