Nuprl Lemma : equal-cubical-identity-at
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[I:Cname List]. ∀[alpha:X(I)]. ∀[p,q:I-path(X;A;a;b;I;alpha)].
p = q ∈ (Id_A a b)(alpha) supposing path-eq(X;A;I;alpha;p;q)
Proof
Definitions occuring in Statement :
cubical-identity: (Id_A a b),
path-eq: path-eq(X;A;I;alpha;p;q),
I-path: I-path(X;A;a;b;I;alpha),
cubical-term: {X ⊢ _:AF},
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
I-cube: X(I),
cubical-set: CubicalSet,
coordinate_name: Cname,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cubical-type-at: A(a),
pi1: fst(t),
cubical-identity: (Id_A a b),
cubical-path: cubical-path(X;A;a;b;I;alpha),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ
Lemmas referenced :
quotient-member-eq,
I-path_wf,
path-eq_wf,
path-eq-equiv,
I-cube_wf,
list_wf,
coordinate_name_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
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{}[p,q:I-path(X;A;a;b;I;alpha)].
p = q supposing path-eq(X;A;I;alpha;p;q)
Date html generated:
2016_06_16-PM-07_31_47
Last ObjectModification:
2015_12_28-PM-04_12_31
Theory : cubical!sets
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