Nuprl Lemma : id-fiber-contraction_wf
∀[X:j⊢]. ∀[T:{X ⊢ _}].
(id-fiber-contraction(X;T) ∈ {X.T.Σ (T)p (Path_((T)p)p (q)p q) ⊢ _
:(Path_(Σ (T)p (Path_((T)p)p (q)p q))p (id-fiber-center(X;T))p q)})
Proof
Definitions occuring in Statement :
id-fiber-contraction: id-fiber-contraction(X;T),
id-fiber-center: id-fiber-center(X;T),
path-type: (Path_A a b),
cubical-sigma: Σ 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,
id-fiber-center: id-fiber-center(X;T),
id-fiber-contraction: id-fiber-contraction(X;T),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
prop: ℙ,
squash: ↓T,
true: True,
cubical-type: {X ⊢ _},
cc-snd: q,
cc-fst: p,
csm-ap-term: (t)s,
csm-ap-type: (AF)s,
csm-id-adjoin: [u],
csm-ap: (s)x,
csm-id: 1(X),
csm-adjoin: (s;u),
pi1: fst(t),
pi2: snd(t),
csm-comp: G o F,
compose: f o g,
uimplies: b supposing a,
guard: {T},
implies: P ⇒ Q,
cube-context-adjoin: X.A,
cubical-pair: cubical-pair(u;v),
sigma-unelim-csm: SigmaUnElim,
cc-adjoin-cube: (v;u),
cubical-term-at: u(a)
Lemmas referenced :
id-fiber-center_wf,
cubical-type_wf,
cubical_set_wf,
sigma-elim-rule,
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-sigma_wf,
csm-path-type,
sigma-unelim-csm_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-id-adjoin_wf,
cubical-term_wf,
csm_id_adjoin_fst_type_lemma,
singleton-contraction_wf,
csm-cubical-sigma,
subtype_rel_universe1,
cubical-type-cumulativity,
csm-adjoin_wf,
csm-comp_wf,
csm-ap-comp-type-sq2,
csm-ap-comp-type-sq,
csm-cubical-pair,
cubical-pair_wf,
csm-ap-comp-term-sq2,
csm-cubical-refl,
csm-ap-comp-term-sq,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
equal_functionality_wrt_subtype_rel2,
I_cube_pair_redex_lemma,
cube_set_restriction_pair_lemma,
cubical-term-at_wf,
sigma-unelim-p-type,
sigma-unelim-p-term
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality,
because_Cache,
dependent_functionElimination,
hyp_replacement,
lambdaEquality_alt,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
setElimination,
rename,
productElimination,
Error :memTop,
applyLambdaEquality,
functionExtensionality,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T:\{X \mvdash{} \_\}].
(id-fiber-contraction(X;T) \mmember{} \{X.T.\mSigma{} (T)p (Path\_((T)p)p (q)p q) \mvdash{} \_
:(Path\_(\mSigma{} (T)p (Path\_((T)p)p (q)p q))p (id-fiber-center(X;T))p q)\})
Date html generated:
2020_05_20-PM-03_31_14
Last ObjectModification:
2020_04_09-AM-10_06_05
Theory : cubical!type!theory
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