Nuprl Lemma : contr-witness_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[c:{X ⊢ _:A}]. ∀[p:{X.A ⊢ _:(Path_(A)p (c)p q)}].
(contr-witness(X;c;p) ∈ {X ⊢ _:Contractible(A)})
Proof
Definitions occuring in Statement :
contr-witness: contr-witness(X;c;p),
contractible-type: Contractible(A),
path-type: (Path_A a b),
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
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,
contr-witness: contr-witness(X;c;p),
contractible-type: Contractible(A),
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
true: True,
all: ∀x:A. B[x],
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cubical-type: {X ⊢ _},
cc-snd: q,
cc-fst: p,
csm-ap-type: (AF)s,
csm-id-adjoin: [u],
csm-comp: G o F,
csm-ap: (s)x,
csm-id: 1(X),
csm-adjoin: (s;u),
compose: f o g,
pi1: fst(t),
csm-ap-term: (t)s,
pi2: snd(t)
Lemmas referenced :
cubical-pair_wf,
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,
cubical-type_wf,
cubical_set_wf,
squash_wf,
true_wf,
cubical-lambda_wf,
equal_wf,
istype-universe,
csm-cubical-pi,
csm-id-adjoin_wf,
subtype_rel_self,
iff_weakening_equal,
csm_id_adjoin_fst_type_lemma,
csm-ap-id-type,
csm-path-type,
csm-adjoin_wf,
csm-comp_wf,
subtype_rel-equal,
cube_set_map_wf,
csm-id_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
universeEquality,
dependent_functionElimination,
independent_isectElimination,
productElimination,
independent_functionElimination,
Error :memTop,
setElimination,
rename
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[c:\{X \mvdash{} \_:A\}]. \mforall{}[p:\{X.A \mvdash{} \_:(Path\_(A)p (c)p q)\}].
(contr-witness(X;c;p) \mmember{} \{X \mvdash{} \_:Contractible(A)\})
Date html generated:
2020_05_20-PM-03_23_30
Last ObjectModification:
2020_04_07-PM-03_49_59
Theory : cubical!type!theory
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