Nuprl Lemma : Kan-cubical-type-equal
∀[X:CubicalSet]. ∀[A:{X ⊢ _(Kan)}]. ∀[B:A:{X ⊢ _} × (I:(Cname List)
⟶ alpha:X(I)
⟶ J:(nameset(I) List)
⟶ x:nameset(I)
⟶ i:ℕ2
⟶ A-open-box(X;A;I;alpha;J;x;i)
⟶ A(alpha))].
A = B ∈ {X ⊢ _(Kan)}
supposing A
= B
∈ (A:{X ⊢ _} × (I:(Cname List)
⟶ alpha:X(I)
⟶ J:(nameset(I) List)
⟶ x:nameset(I)
⟶ i:ℕ2
⟶ A-open-box(X;A;I;alpha;J;x;i)
⟶ A(alpha)))
Proof
Definitions occuring in Statement :
Kan-cubical-type: {X ⊢ _(Kan)},
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
I-cube: X(I),
cubical-set: CubicalSet,
nameset: nameset(L),
coordinate_name: Cname,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
Kan-cubical-type: {X ⊢ _(Kan)},
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
nameset: nameset(L)
Lemmas referenced :
and_wf,
Kan-A-filler_wf,
uniform-Kan-A-filler_wf,
equal_wf,
cubical-type_wf,
list_wf,
coordinate_name_wf,
I-cube_wf,
nameset_wf,
int_seg_wf,
A-open-box_wf,
subtype_rel_list,
cubical-type-at_wf,
Kan-cubical-type_wf,
cubical-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
hypothesis,
spreadEquality,
hypothesisEquality,
lemma_by_obid,
isectElimination,
instantiate,
productEquality,
functionEquality,
applyEquality,
lambdaEquality,
cumulativity,
universeEquality,
sqequalRule,
because_Cache,
natural_numberEquality,
dependent_functionElimination,
independent_isectElimination,
productElimination,
dependent_pairEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_(Kan)\}]. \mforall{}[B:A:\{X \mvdash{} \_\} \mtimes{} (I:(Cname List)
{}\mrightarrow{} alpha:X(I)
{}\mrightarrow{} J:(nameset(I) List)
{}\mrightarrow{} x:nameset(I)
{}\mrightarrow{} i:\mBbbN{}2
{}\mrightarrow{} A-open-box(X;A;I;alpha;J;x;i)
{}\mrightarrow{} A(alpha))].
A = B supposing A = B
Date html generated:
2016_06_16-PM-06_44_35
Last ObjectModification:
2015_12_28-PM-04_25_26
Theory : cubical!sets
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