Nuprl Lemma : csm-contractible-type
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[Z:j⊢]. ∀[s:Z j⟶ X]. ((Contractible(A))s = Z ⊢ Contractible((A)s) ∈ {Z ⊢ _})
Proof
Definitions occuring in Statement :
contractible-type: Contractible(A),
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
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,
subtype_rel: A ⊆r B,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
all: ∀x:A. B[x],
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g,
prop: ℙ,
squash: ↓T,
true: True,
and: P ∧ Q,
uimplies: b supposing a,
contractible-type: Contractible(A),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cubical-type: {X ⊢ _},
cc-snd: q,
csm-ap-type: (AF)s,
cc-fst: p,
csm-comp: G o F,
csm-ap: (s)x,
csm-adjoin: (s;u),
csm-ap-term: (t)s
Lemmas referenced :
cube_set_map_wf,
cubical-type_wf,
cubical_set_wf,
csm-ap-comp-type,
cubical_set_cumulativity-i-j,
cube-context-adjoin_wf,
csm-ap-type_wf,
cubical-type-cumulativity2,
cc-fst_wf,
subtype_rel_self,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-comp_wf,
csm-ap-type-fst-adjoin,
subtype_rel-equal,
csm-cubical-sigma,
cubical-pi_wf,
path-type_wf,
csm-ap-term_wf,
cc-snd_wf,
cubical-sigma_wf,
iff_weakening_equal,
csm-cubical-pi,
csm-adjoin_wf,
csm-adjoin-comp,
csm-comp-type,
cubical-term_wf,
csm-path-type,
csm-ap-term-snd-adjoin
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality,
because_Cache,
applyLambdaEquality,
equalitySymmetry,
hyp_replacement,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
Error :memTop,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
setElimination,
rename,
productElimination,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[Z:j\mvdash{}]. \mforall{}[s:Z j{}\mrightarrow{} X]. ((Contractible(A))s = Z \mvdash{} Contractible((A)s))
Date html generated:
2020_05_20-PM-03_22_52
Last ObjectModification:
2020_04_07-PM-05_05_19
Theory : cubical!type!theory
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