Nuprl Lemma : app-univ-a-1
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}].
(app(UA; f) = (EquivPath(G.Equiv(decode(A);decode(B));(A)p;(B)p;q))[f] ∈ {G ⊢ _:(Path_c𝕌 A B)})
Proof
Definitions occuring in Statement :
univ-a: UA,
equiv-path: EquivPath(G;A;B;f),
universe-decode: decode(t),
cubical-universe: c𝕌,
cubical-equiv: Equiv(T;A),
path-type: (Path_A a b),
cubical-app: app(w; u),
csm-id-adjoin: [u],
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
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,
guard: {T},
univ-a: UA,
all: ∀x:A. B[x],
uimplies: b supposing a,
cubical-lam: cubical-lam(X;b),
csm-id: 1(X),
csm-ap-term: (t)s,
cc-fst: p,
csm-id-adjoin: [u],
csm-ap: (s)x,
csm-adjoin: (s;u),
pi1: fst(t),
squash: ↓T,
prop: ℙ,
true: True
Lemmas referenced :
csm-ap-term-universe,
cubical_set_cumulativity-i-j,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cc-fst_wf,
cubical-lam_wf,
istype-cubical-term,
cubical-equiv_wf,
universe-decode_wf,
istype-cubical-universe-term,
cubical_set_wf,
equiv-path_wf,
cc-snd_wf,
cubical-term-eqcd,
cubical-beta,
cubical-type-cumulativity,
csm-cubical-equiv,
csm-universe-decode,
path-type_wf,
cubical-universe_wf,
csm-path-type,
csm-id-adjoin_wf,
csm-cubical-universe,
squash_wf,
true_wf,
cubical-type_wf,
csm-ap-id-term
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
because_Cache,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
universeIsType,
independent_isectElimination,
lambdaEquality_alt,
hyp_replacement,
Error :memTop,
imageElimination,
inhabitedIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(decode(A);decode(B))\}].
(app(UA; f) = (EquivPath(G.Equiv(decode(A);decode(B));(A)p;(B)p;q))[f])
Date html generated:
2020_05_20-PM-07_31_19
Last ObjectModification:
2020_04_28-PM-11_57_30
Theory : cubical!type!theory
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