Nuprl Lemma : path-eq-same-name
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[I:Cname List]. ∀[alpha:X(I)].
∀p,q:I-path(X;A;a;b;I;alpha).
path-eq(X;A;I;alpha;p;q) ⇐⇒ p = q ∈ I-path(X;A;a;b;I;alpha) supposing (fst(p)) = (fst(q)) ∈ Cname
Proof
Definitions occuring in Statement :
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: {X ⊢ _},
I-cube: X(I),
cubical-set: CubicalSet,
coordinate_name: Cname,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
I-path: I-path(X;A;a;b;I;alpha),
top: Top,
path-eq: path-eq(X;A;I;alpha;p;q),
pi1: fst(t),
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
not: ¬A,
false: False,
squash: ↓T,
true: True,
named-path: named-path(X;A;a;b;I;alpha;z),
cand: A c∧ B
Lemmas referenced :
path-eq_wf,
equal_wf,
I-path_wf,
coordinate_name_wf,
pi1_wf_top,
I-cube_wf,
list_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
equal-named-paths,
l_member_wf,
squash_wf,
true_wf,
named-path_wf,
cubical-type-ap-morph-id,
cons_wf,
rename-one-name_wf,
cube-set-restriction_wf,
iota_wf,
rename-one-same,
path-eq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
productElimination,
independent_pairEquality,
setElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
promote_hyp,
instantiate,
cumulativity,
independent_isectElimination,
intEquality,
lambdaEquality,
natural_numberEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
dependent_pairEquality,
addLevel,
hyp_replacement,
applyEquality,
imageElimination,
universeEquality,
equalityUniverse,
levelHypothesis,
imageMemberEquality,
baseClosed
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). path-eq(X;A;I;alpha;p;q) \mLeftarrow{}{}\mRightarrow{} p = q supposing (fst(p)) = (fst(q))
Date html generated:
2016_10_28-PM-00_14_15
Last ObjectModification:
2016_07_12-PM-00_41_05
Theory : cubical!sets
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