Nuprl Lemma : I-path-morph_wf2
∀X:CubicalSet. ∀A:{X ⊢ _}. ∀a,b:{X ⊢ _:A}. ∀I,K:Cname List. ∀f:name-morph(I;K). ∀alpha:X(I).
∀w:cubical-path(X;A;a;b;I;alpha).
(I-path-morph(X;A;I;K;f;alpha;w) ∈ cubical-path(X;A;a;b;K;f(alpha)))
Proof
Definitions occuring in Statement :
cubical-path: cubical-path(X;A;a;b;I;alpha),
I-path-morph: I-path-morph(X;A;I;K;f;alpha;p),
cubical-term: {X ⊢ _:AF},
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,
all: ∀x:A. B[x],
member: t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
cubical-path: cubical-path(X;A;a;b;I;alpha),
member: t ∈ T,
uall: ∀[x:A]. B[x],
quotient: x,y:A//B[x; y],
and: P ∧ Q,
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
prop: ℙ
Lemmas referenced :
cubical-path_wf,
cube-set-restriction_wf,
I-path_wf,
quotient-member-eq,
path-eq_wf,
path-eq-equiv,
I-path-morph_wf,
I-path-morph_functionality,
equal_wf,
equal-wf-base,
I-cube_wf,
name-morph_wf,
list_wf,
coordinate_name_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
rename,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
productEquality
Latex:
\mforall{}X:CubicalSet. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}a,b:\{X \mvdash{} \_:A\}. \mforall{}I,K:Cname List. \mforall{}f:name-morph(I;K). \mforall{}alpha:X(I).
\mforall{}w:cubical-path(X;A;a;b;I;alpha).
(I-path-morph(X;A;I;K;f;alpha;w) \mmember{} cubical-path(X;A;a;b;K;f(alpha)))
Date html generated:
2018_05_23-PM-06_55_57
Last ObjectModification:
2018_05_18-AM-08_31_38
Theory : cubical!sets
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