Nuprl Lemma : equal-cubical-equiv-at
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[x:Equiv(T;A)(a)]. ∀[y:u:(T ⟶ A)(a) × IsEquiv((T)p;(A)p;q)((a;u))].
x = y ∈ Equiv(T;A)(a) supposing x = y ∈ (u:(T ⟶ A)(a) × IsEquiv((T)p;(A)p;q)((a;u)))
Proof
Definitions occuring in Statement :
cubical-equiv: Equiv(T;A),
is-cubical-equiv: IsEquiv(T;A;w),
cubical-fun: (A ⟶ B),
cc-snd: q,
cc-fst: p,
cc-adjoin-cube: (v;u),
cube-context-adjoin: X.A,
csm-ap-type: (AF)s,
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
product: x:A × B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
cubical-equiv: Equiv(T;A),
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B
Lemmas referenced :
cubical-sigma-at,
istype-cubical-type-at,
cubical-equiv_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
I_cube_wf,
fset_wf,
nat_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
Error :memTop,
hypothesis,
sqequalRule,
equalityIstype,
inhabitedIsType,
hypothesisEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
instantiate,
applyEquality,
universeIsType,
because_Cache
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T,A:\{X \mvdash{} \_\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[a:X(I)]. \mforall{}[x:Equiv(T;A)(a)].
\mforall{}[y:u:(T {}\mrightarrow{} A)(a) \mtimes{} IsEquiv((T)p;(A)p;q)((a;u))].
x = y supposing x = y
Date html generated:
2020_05_20-PM-03_26_45
Last ObjectModification:
2020_04_06-PM-06_45_00
Theory : cubical!type!theory
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