Nuprl Lemma : cubical-refl_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. (refl(a) ∈ {X ⊢ _:(Id_A a a)})
Proof
Definitions occuring in Statement :
cubical-refl: refl(a),
cubical-identity: (Id_A a b),
cubical-term: {X ⊢ _:AF},
cubical-type: {X ⊢ _},
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-refl: refl(a),
cubical-term: {X ⊢ _:AF},
cubical-identity: (Id_A a b),
pi1: fst(t),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
cubical-path: cubical-path(X;A;a;b;I;alpha),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
quotient: x,y:A//B[x; y],
implies: P ⇒ Q,
refl-path: refl-path(A;a;I;alpha),
I-path-morph: I-path-morph(X;A;I;K;f;alpha;p),
path-eq: path-eq(X;A;I;alpha;p;q),
iota': iota'(I),
add-fresh-cname: I+,
named-path-morph: named-path-morph(X;A;I;K;z;x;f;alpha;w),
has-value: (a)↓,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
cand: A c∧ B,
not: ¬A,
false: False,
cubical-type-at: A(a),
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
pi2: snd(t),
cubical-type-ap-morph: (u a f),
cubical-type: {X ⊢ _}
Lemmas referenced :
refl-path_wf,
subtype_quotient,
I-path_wf,
path-eq_wf,
path-eq-equiv,
I-cube_wf,
list_wf,
coordinate_name_wf,
name-morph_wf,
cubical-path_wf,
cube-set-restriction_wf,
I-path-morph_wf2,
subtype_rel_self,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf,
quotient-member-eq,
I-path-morph_wf,
value-type-has-value,
not_wf,
l_member_wf,
set-value-type,
coordinate_name-value-type,
fresh-cname_wf,
cubical-type-ap-morph-id,
cons_wf,
rename-one-name_wf,
iota_wf,
cubical-type-ap-morph_wf,
extend-name-morph_wf,
cubical-type-at_wf,
rename-one-same,
id-morph_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
iff_weakening_equal,
extend-name-morph-iota,
name-comp_wf,
cube-set-restriction-comp,
cubical-type-ap-morph-comp,
subtype_rel-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
isectElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
universeIsType,
lambdaFormation_alt,
functionIsType,
because_Cache,
equalityIstype,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
independent_functionElimination,
rename,
callbyvalueReduce,
setEquality,
setElimination,
voidElimination,
independent_pairFormation,
natural_numberEquality,
imageElimination,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
hyp_replacement
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a:\{X \mvdash{} \_:A\}]. (refl(a) \mmember{} \{X \mvdash{} \_:(Id\_A a a)\})
Date html generated:
2020_05_21-AM-11_14_22
Last ObjectModification:
2019_12_10-PM-00_06_11
Theory : cubical!sets
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