Nuprl Lemma : path-eq_wf
∀[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)
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,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
I-path: I-path(X;A;a;b;I;alpha),
named-path: named-path(X;A;a;b;I;alpha;z),
path-eq: path-eq(X;A;I;alpha;p;q),
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True
Lemmas referenced :
equal_wf,
cubical-type-at_wf,
cons_wf,
coordinate_name_wf,
cube-set-restriction_wf,
iota_wf,
squash_wf,
true_wf,
I-cube_wf,
cube-set-restriction-comp,
rename-one-name_wf,
iff_weakening_equal,
rename-one-iota,
cubical-type-ap-morph_wf,
I-path_wf,
list_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
setElimination,
rename,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
imageElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
independent_isectElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
independent_functionElimination,
natural_numberEquality,
hyp_replacement
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) \mmember{} \mBbbP{})
Date html generated:
2017_10_05-PM-03_54_46
Last ObjectModification:
2017_07_28-AM-11_27_49
Theory : cubical!sets
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