Nuprl Lemma : is-prop-contractible
∀X:j⊢. ∀A:{X ⊢ _}. ({X ⊢ _:isProp(A)} ⇒ {X ⊢ _:A} ⇒ {X ⊢ _:Contractible(A)})
Proof
Definitions occuring in Statement :
is-prop: isProp(A),
contractible-type: Contractible(A),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
is-prop: isProp(A),
contractible-type: Contractible(A),
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B
Lemmas referenced :
cubical-pi-implies-sigma,
cubical-pi_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
csm-ap-type_wf,
cc-fst_wf,
path-type_wf,
csm-ap-term_wf,
cc-snd_wf,
cubical-term_wf,
is-prop_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
hypothesisEquality,
instantiate,
isectElimination,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
independent_functionElimination,
universeIsType
Latex:
\mforall{}X:j\mvdash{}. \mforall{}A:\{X \mvdash{} \_\}. (\{X \mvdash{} \_:isProp(A)\} {}\mRightarrow{} \{X \mvdash{} \_:A\} {}\mRightarrow{} \{X \mvdash{} \_:Contractible(A)\})
Date html generated:
2020_05_20-PM-03_35_25
Last ObjectModification:
2020_04_06-PM-07_00_53
Theory : cubical!type!theory
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