Nuprl Lemma : extend-to-contraction_wf
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}. ∀ext:Gamma +⊢ Extension(A).
(extend-to-contraction(Gamma;A;ext) ∈ {Gamma ⊢ _:Contractible(A)})
Proof
Definitions occuring in Statement :
extend-to-contraction: extend-to-contraction(Gamma;A;ext),
uniform-extend: uniform-extend{i:l}(Gamma; A),
contractible-type: Contractible(A),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
all: ∀x:A. B[x],
member: t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uniform-extend: uniform-extend{i:l}(Gamma; A),
extension-fun: extension-fun{i:l}(Gamma;A),
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
extend-to-contraction: extend-to-contraction(Gamma;A;ext),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
cubical-type: {X ⊢ _},
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
csm-comp: G o F,
compose: f o g,
csm-ap: (s)x,
squash: ↓T,
prop: ℙ,
true: True,
uniform-extension-fun: uniform-extension-fun{i:l}(Gamma;A;ext),
pi1: fst(t),
csm-adjoin: (s;u),
csm-id: 1(X),
csm-id-adjoin: [u],
interval-0: 0(𝕀),
implies: P ⇒ Q,
interval-1: 1(𝕀),
csm-ap-term: (t)s,
pi2: snd(t),
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
csm-ap-term_wf,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
interval-type_wf,
csm-ap-type_wf,
cc-fst_wf,
cc-fst_wf_interval,
cc-snd_wf,
contr-witness_wf,
csm-id_wf,
face-0_wf,
empty-context-subset-lemma3,
subtype_rel_set,
cubical-term-eqcd,
equal-wf-base-T,
thin-context-subset,
context-subset_wf,
istype-cubical-term,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-id-type,
term-to-path_wf,
csm-comp_wf,
face-one_wf,
context-subset-is-subset,
context-subset-term-subtype,
path-type_wf,
squash_wf,
true_wf,
uniform-extend_wf,
cubical-type_wf,
cubical_set_wf,
csm-id-adjoin_wf-interval-0,
istype-universe,
equal_wf,
csm-face-0,
face-one-interval-0,
csm-interval-0,
cc_snd_csm_id_adjoin_lemma,
csm-face-one,
csm_id_adjoin_fst_term_lemma,
context-subset-map,
csm-id-adjoin_wf-interval-1,
interval-1_wf,
subset-cubical-term,
context-1-subset,
face-one-interval-1,
sub_cubical_set_wf,
face-type_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
instantiate,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality_alt,
baseClosed,
dependent_functionElimination,
productElimination,
imageElimination,
universeIsType,
inhabitedIsType,
natural_numberEquality,
imageMemberEquality,
hyp_replacement,
universeEquality,
applyLambdaEquality,
equalityIstype,
independent_functionElimination
Latex:
\mforall{}Gamma:j\mvdash{}. \mforall{}A:\{Gamma \mvdash{} \_\}. \mforall{}ext:Gamma +\mvdash{} Extension(A).
(extend-to-contraction(Gamma;A;ext) \mmember{} \{Gamma \mvdash{} \_:Contractible(A)\})
Date html generated:
2020_05_20-PM-05_22_45
Last ObjectModification:
2020_05_02-PM-03_25_05
Theory : cubical!type!theory
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