Nuprl Lemma : equal-named-paths
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[I:Cname List]. ∀[alpha:X(I)]. ∀[z:Cname].
∀[p:named-path(X;A;a;b;I;alpha;z)]. ∀[q:A(iota(z)(alpha))].
p = q ∈ named-path(X;A;a;b;I;alpha;z) supposing p = q ∈ A(iota(z)(alpha))
supposing ¬(z ∈ I)
Proof
Definitions occuring in Statement :
named-path: named-path(X;A;a;b;I;alpha;z),
cubical-term: {X ⊢ _:AF},
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
iota: iota(x),
coordinate_name: Cname,
l_member: (x ∈ l),
cons: [a / b],
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
named-path: named-path(X;A;a;b;I;alpha;z),
prop: ℙ
Lemmas referenced :
name-path-endpoints_wf,
equal_wf,
cubical-type-at_wf,
cons_wf,
coordinate_name_wf,
cube-set-restriction_wf,
iota_wf,
named-path_wf,
not_wf,
l_member_wf,
I-cube_wf,
list_wf,
cubical-term_wf,
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,
lemma_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}]. \mforall{}[I:Cname List]. \mforall{}[alpha:X(I)]. \mforall{}[z:Cname].
\mforall{}[p:named-path(X;A;a;b;I;alpha;z)]. \mforall{}[q:A(iota(z)(alpha))]. p = q supposing p = q
supposing \mneg{}(z \mmember{} I)
Date html generated:
2016_06_16-PM-07_28_02
Last ObjectModification:
2015_12_28-PM-04_14_29
Theory : cubical!sets
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