Nuprl Lemma : name-path-endpoints_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[I:Cname List]. ∀[alpha:X(I)]. ∀[z:Cname]. ∀[omega:A(iota(z)(alpha))].
name-path-endpoints(X;A;a;b;I;alpha;z;omega) ∈ ℙ supposing ¬(z ∈ I)
Proof
Definitions occuring in Statement :
name-path-endpoints: name-path-endpoints(X;A;a;b;I;alpha;z;omega),
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],
prop: ℙ,
not: ¬A,
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
name-path-endpoints: name-path-endpoints(X;A;a;b;I;alpha;z;omega),
prop: ℙ,
and: P ∧ Q,
squash: ↓T,
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
less_than: a < b,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cubical-type-at: A(a),
cubical-term: {X ⊢ _:AF}
Lemmas referenced :
equal_wf,
cubical-type-at_wf,
I-cube_wf,
cube-set-restriction-comp,
cons_wf,
coordinate_name_wf,
iota_wf,
face-map_wf,
false_wf,
lelt_wf,
iff_weakening_equal,
cubical-type-ap-morph_wf,
cube-set-restriction_wf,
iota-identity,
not_wf,
l_member_wf,
list_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf,
cube-set-restriction-id,
squash_wf,
true_wf,
name-morph_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
productEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
imageElimination,
because_Cache,
dependent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
lambdaFormation,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
independent_functionElimination,
hyp_replacement,
setElimination,
rename,
axiomEquality,
isect_memberEquality
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{}[omega:A(iota(z)(alpha))].
name-path-endpoints(X;A;a;b;I;alpha;z;omega) \mmember{} \mBbbP{} supposing \mneg{}(z \mmember{} I)
Date html generated:
2017_10_05-PM-03_54_13
Last ObjectModification:
2017_07_28-AM-11_27_34
Theory : cubical!sets
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