Nuprl Lemma : cubical-fiber-id-fun
∀X:j⊢. ∀T:{X ⊢ _}. ∀[u:{X ⊢ _:T}]. (X ⊢ Fiber(cubical-id-fun(X);u) = Σ T (Path_(T)p (u)p q) ∈ {X ⊢ _})
Proof
Definitions occuring in Statement :
cubical-fiber: Fiber(w;a),
path-type: (Path_A a b),
cubical-sigma: Σ A B,
cubical-id-fun: cubical-id-fun(X),
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],
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
cubical-fiber: Fiber(w;a),
member: t ∈ T,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True
Lemmas referenced :
cubical-sigma_wf,
squash_wf,
true_wf,
cubical-type_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
path-type_wf,
csm-ap-type_wf,
cc-fst_wf,
csm-ap-term_wf,
cubical-term_wf,
cubical_set_wf,
csm-cubical-id-fun,
cubical-app_wf_fun,
cc-snd_wf,
equal_wf,
istype-universe,
cubical-app-id-fun
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
instantiate,
sqequalRule,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
hyp_replacement,
universeEquality
Latex:
\mforall{}X:j\mvdash{}. \mforall{}T:\{X \mvdash{} \_\}. \mforall{}[u:\{X \mvdash{} \_:T\}]. (X \mvdash{} Fiber(cubical-id-fun(X);u) = \mSigma{} T (Path\_(T)p (u)p q))
Date html generated:
2020_05_20-PM-03_27_34
Last ObjectModification:
2020_04_07-PM-05_21_46
Theory : cubical!type!theory
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