Nuprl Lemma : cubical-universe-at-equal
∀[X:j⊢]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[x,y:c𝕌(a)].
x = y ∈ c𝕌(a)
supposing ((fst(x)) = (fst(y)) ∈ {formal-cube(I) ⊢ _}) ∧ ((snd(x)) = (snd(y)) ∈ formal-cube(I) ⊢ CompOp(fst(x)))
Proof
Definitions occuring in Statement :
cubical-universe: c𝕌,
composition-op: Gamma ⊢ CompOp(A),
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
formal-cube: formal-cube(I),
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
cubical-universe-at,
composition-op_wf,
formal-cube_wf1,
cubical-type-cumulativity2,
subtype_rel-equal,
cubical-type_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
because_Cache,
productElimination,
dependent_pairEquality_alt,
universeIsType,
instantiate,
hypothesisEquality,
applyEquality,
productIsType,
equalityIstype,
inhabitedIsType,
independent_isectElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[a:X(I)]. \mforall{}[x,y:c\mBbbU{}(a)].
x = y supposing ((fst(x)) = (fst(y))) \mwedge{} ((snd(x)) = (snd(y)))
Date html generated:
2020_05_20-PM-07_07_59
Last ObjectModification:
2020_04_25-PM-03_20_12
Theory : cubical!type!theory
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