Nuprl Lemma : cubical-type-equal
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:A:I:(Cname List) ⟶ X(I) ⟶ Type × (I:(Cname List)
⟶ J:(Cname List)
⟶ f:name-morph(I;J)
⟶ a:X(I)
⟶ (A I a)
⟶ (A J f(a)))].
A = B ∈ {X ⊢ _}
supposing A
= B
∈ (A:I:(Cname List) ⟶ X(I) ⟶ Type × (I:(Cname List)
⟶ J:(Cname List)
⟶ f:name-morph(I;J)
⟶ a:X(I)
⟶ (A I a)
⟶ (A J f(a))))
Proof
Definitions occuring in Statement :
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
name-morph: name-morph(I;J),
coordinate_name: Cname,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cubical-type: {X ⊢ _},
and: P ∧ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_apply: x[s],
all: ∀x:A. B[x],
prop: ℙ
Lemmas referenced :
all_wf,
list_wf,
coordinate_name_wf,
I-cube_wf,
equal_wf,
id-morph_wf,
subtype_rel-equal,
cube-set-restriction_wf,
cube-set-restriction-id,
iff_weakening_equal,
name-morph_wf,
name-comp_wf,
cube-set-restriction-comp,
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,
productElimination,
sqequalRule,
productEquality,
extract_by_obid,
isectElimination,
lambdaEquality,
hypothesisEquality,
applyEquality,
functionExtensionality,
because_Cache,
independent_isectElimination,
instantiate,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_functionElimination,
functionEquality,
cumulativity,
dependent_pairEquality,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:A:I:(Cname List) {}\mrightarrow{} X(I) {}\mrightarrow{} Type \mtimes{} (I:(Cname List)
{}\mrightarrow{} J:(Cname List)
{}\mrightarrow{} f:name-morph(I;J)
{}\mrightarrow{} a:X(I)
{}\mrightarrow{} (A I a)
{}\mrightarrow{} (A J f(a)))].
A = B supposing A = B
Date html generated:
2017_10_05-AM-10_12_43
Last ObjectModification:
2017_07_28-AM-11_18_28
Theory : cubical!sets
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