Nuprl Lemma : path-eq-equiv
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[I:Cname List]. ∀[alpha:X(I)].
EquivRel(I-path(X;A;a;b;I;alpha);p,q.path-eq(X;A;I;alpha;p;q))
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,
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
I-path: I-path(X;A;a;b;I;alpha),
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
path-eq: path-eq(X;A;I;alpha;p;q),
member: t ∈ T,
uimplies: b supposing a,
cand: A c∧ B,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
false: False,
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bnot: ¬bb,
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
guard: {T},
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
int_upper: {i...},
coordinate_name: Cname,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
nameset: nameset(L),
rename-one-name: rename-one-name(z1;z2),
id-morph: 1,
name-morph: name-morph(I;J),
subtype_rel: A ⊆r B,
squash: ↓T,
named-path: named-path(X;A;a;b;I;alpha;z),
true: True
Lemmas referenced :
cubical-type-ap-morph-id,
cons_wf,
coordinate_name_wf,
rename-one-name_wf,
cube-set-restriction_wf,
iota_wf,
l_member_wf,
istype-void,
named-path_wf,
path-eq_wf,
I-cube_wf,
list_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf,
nameset_wf,
equal-wf-T-base,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_wf,
bool_cases_sqequal,
bool_subtype_base,
eqff_to_assert,
nameset_subtype_extd-nameset,
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
subtype_base_sq,
assert-eq-cname,
eqtt_to_assert,
eq-cname_wf,
id-morph_wf,
name-morphs-equal,
equal_wf,
cubical-type-at_wf,
cube-set-restriction-comp,
iff_weakening_equal,
cubical-type-ap-morph-comp,
rename-one-same,
name-morph_wf,
cube-set-restriction-when-id,
rename-one-comp,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
rename-one-iota,
cubical-type-ap-morph_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
independent_pairFormation,
lambdaFormation_alt,
productElimination,
thin,
setElimination,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
because_Cache,
independent_isectElimination,
productIsType,
setIsType,
universeIsType,
sqequalRule,
functionIsType,
independent_functionElimination,
voidElimination,
equalityIsType1,
promote_hyp,
equalityIsType3,
dependent_pairFormation_alt,
dependent_functionElimination,
natural_numberEquality,
closedConclusion,
intEquality,
cumulativity,
instantiate,
equalityElimination,
unionElimination,
functionExtensionality,
equalitySymmetry,
equalityTransitivity,
inhabitedIsType,
lambdaEquality_alt,
applyEquality,
dependent_set_memberEquality_alt,
equalityIstype,
hyp_replacement,
applyLambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}]. \mforall{}[I:Cname List]. \mforall{}[alpha:X(I)].
EquivRel(I-path(X;A;a;b;I;alpha);p,q.path-eq(X;A;I;alpha;p;q))
Date html generated:
2019_11_06-PM-00_38_59
Last ObjectModification:
2018_12_13-PM-03_03_37
Theory : cubical!sets
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