Nuprl Lemma : path-contraction_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[pth:{X ⊢ _:(Path_A a b)}].
(path-contraction(X;pth) ∈ {X.𝕀 ⊢ _:(Path_(A)p (a)p path-point(pth))})
Proof
Definitions occuring in Statement :
path-contraction: path-contraction(X;pth),
path-point: path-point(pth),
path-type: (Path_A a b),
interval-type: 𝕀,
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,
path-contraction: path-contraction(X;pth),
guard: {T},
subtype_rel: A ⊆r B,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
squash: ↓T,
prop: ℙ,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True,
all: ∀x:A. B[x],
csm-ap-term: (t)s,
csm-id: 1(X),
csm-ap: (s)x,
interval-0: 0(𝕀),
interval-1: 1(𝕀),
path-point: path-point(pth)
Lemmas referenced :
cubical-path-app_wf,
cube-context-adjoin_wf,
interval-type_wf,
cubical_set_cumulativity-i-j,
csm-ap-type_wf,
cc-fst_wf,
csm-ap-term_wf,
interval-meet_wf,
csm-interval-type,
cc-snd_wf,
cubical-term_wf,
path-type_wf,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
path-type-p,
subtype_rel_self,
iff_weakening_equal,
cube_set_map_wf,
term-to-path_wf,
csm_id_adjoin_fst_term_lemma,
csm-cubical-path-app,
cc_snd_csm_id_adjoin_lemma,
csm-interval-meet,
interval-meet-0,
csm-path-type,
cubical-path-app-0,
interval-meet-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
instantiate,
hypothesis,
hypothesisEquality,
applyEquality,
sqequalRule,
because_Cache,
equalityTransitivity,
equalitySymmetry,
Error :memTop,
axiomEquality,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
lambdaEquality_alt,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
natural_numberEquality,
hyp_replacement,
dependent_functionElimination,
applyLambdaEquality,
lambdaFormation_alt,
equalityIstype
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}]. \mforall{}[pth:\{X \mvdash{} \_:(Path\_A a b)\}].
(path-contraction(X;pth) \mmember{} \{X.\mBbbI{} \mvdash{} \_:(Path\_(A)p (a)p path-point(pth))\})
Date html generated:
2020_05_20-PM-03_28_22
Last ObjectModification:
2020_04_07-PM-05_30_01
Theory : cubical!type!theory
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