Nuprl Lemma : refl-path_wf
∀X:CubicalSet. ∀A:{X ⊢ _}. ∀a:{X ⊢ _:A}. ∀I:Cname List. ∀alpha:X(I). (refl-path(A;a;I;alpha) ∈ I-path(X;A;a;a;I;alpha))
Proof
Definitions occuring in Statement :
refl-path: refl-path(A;a;I;alpha),
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,
all: ∀x:A. B[x],
member: t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
refl-path: refl-path(A;a;I;alpha),
I-path: I-path(X;A;a;b;I;alpha),
uall: ∀[x:A]. B[x],
named-path: named-path(X;A;a;b;I;alpha;z),
cubical-term: {X ⊢ _:AF},
subtype_rel: A ⊆r B,
cubical-type-at: A(a),
uimplies: b supposing a,
prop: ℙ,
not: ¬A,
implies: P ⇒ Q,
false: False,
add-fresh-cname: I+,
has-value: (a)↓,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
iota: iota(x),
iota': iota'(I),
true: True,
name-path-endpoints: name-path-endpoints(X;A;a;b;I;alpha;z;omega),
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
fresh-cname_wf,
cubical-type-ap-morph_wf,
add-fresh-cname_wf,
iota'_wf,
subtype_rel_self,
cubical-type-at_wf,
name-path-endpoints_wf,
fresh-cname-not-member2,
named-path_wf,
l_member_wf,
coordinate_name_wf,
I-cube_wf,
list_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf,
value-type-has-value,
not_wf,
set-value-type,
coordinate_name-value-type,
cube-set-restriction_wf,
face-map_wf_fresh-cname,
istype-false,
istype-le,
istype-less_than,
equal_wf,
cubical-type-ap-morph-id,
name-comp_wf,
iota'-identity,
iff_weakening_equal,
squash_wf,
true_wf,
istype-universe,
cubical-type-ap-morph-comp,
subtype_rel-equal,
cube-set-restriction-id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
dependent_pairEquality_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality_alt,
applyEquality,
setElimination,
rename,
because_Cache,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
hyp_replacement,
universeIsType,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
callbyvalueReduce,
setEquality,
imageElimination,
imageMemberEquality,
baseClosed,
natural_numberEquality,
independent_pairFormation,
productIsType,
productElimination,
inhabitedIsType,
instantiate,
universeEquality
Latex:
\mforall{}X:CubicalSet. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}a:\{X \mvdash{} \_:A\}. \mforall{}I:Cname List. \mforall{}alpha:X(I).
(refl-path(A;a;I;alpha) \mmember{} I-path(X;A;a;a;I;alpha))
Date html generated:
2019_11_06-PM-00_53_38
Last ObjectModification:
2018_11_10-PM-03_16_57
Theory : cubical!sets
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