Nuprl Lemma : csm-cubical-identity
∀X,Delta:CubicalSet. ∀s:Delta ⟶ X. ∀A:{X ⊢ _}. ∀a,b:{X ⊢ _:A}.
(((Id_A a b))s
= (Id_(A)s (a)s (b)s)
∈ (A:I:(Cname List) ⟶ Delta(I) ⟶ Type × (I:(Cname List)
⟶ J:(Cname List)
⟶ f:name-morph(I;J)
⟶ a:Delta(I)
⟶ (A I a)
⟶ (A J f(a)))))
Proof
Definitions occuring in Statement :
cubical-identity: (Id_A a b),
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:AF},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I-cube: X(I),
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
name-morph: name-morph(I;J),
coordinate_name: Cname,
list: T List,
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
cubical-identity: (Id_A a b),
cubical-path: cubical-path(X;A;a;b;I;alpha),
squash: ↓T,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
and: P ∧ Q,
I-path: I-path(X;A;a;b;I;alpha),
path-eq: path-eq(X;A;I;alpha;p;q),
cubical-type-at: A(a),
pi1: fst(t),
top: Top,
cand: A c∧ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
named-path: named-path(X;A;a;b;I;alpha;z),
quotient: x,y:A//B[x; y],
I-path-morph: I-path-morph(X;A;I;K;f;alpha;p),
named-path-morph: named-path-morph(X;A;I;K;z;x;f;alpha;w)
Lemmas referenced :
csm-ap-type_wf,
I-cube_wf,
list_wf,
coordinate_name_wf,
cubical-path_wf,
csm-ap_wf,
name-morph_wf,
cube-set-restriction_wf,
cubical-term_wf,
cubical-type_wf,
cube-set-map_wf,
cubical-set_wf,
csm-I-path,
quotient_wf,
squash_wf,
equiv_rel_wf,
path-eq-equiv,
equal_wf,
true_wf,
cons_wf,
csm-ap-restriction,
iota_wf,
csm-cubical-type-ap-morph,
cubical-type-ap-morph_wf,
rename-one-name_wf,
l_member_wf,
subtype_rel_self,
iff_weakening_equal,
cube-set-restriction-comp,
rename-one-iota,
cubical-type-at_wf,
subtype_rel-equal,
and_wf,
path-eq_wf,
I-path_wf,
fresh-cname_wf,
named-path_wf,
equal-named-paths,
fresh-cname-not-member2,
named-path-morph_wf,
istype-universe,
extend-name-morph_wf,
cubical-type-ap-morph-comp,
not_wf,
name-comp_wf,
extend-name-morph-rename-one,
extend-name-morph-iota,
istype-void,
csm-type-at,
subtype_quotient
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
setElimination,
rename,
productElimination,
sqequalRule,
dependent_pairEquality_alt,
lambdaEquality_alt,
universeIsType,
because_Cache,
functionIsType,
applyEquality,
inhabitedIsType,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
imageElimination,
independent_isectElimination,
functionEquality,
cumulativity,
universeEquality,
imageMemberEquality,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
independent_functionElimination,
independent_pairFormation,
natural_numberEquality,
instantiate,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
promote_hyp,
equalityIstype,
productIsType,
sqequalBase,
setEquality,
isect_memberEquality_alt,
closedConclusion
Latex:
\mforall{}X,Delta:CubicalSet. \mforall{}s:Delta {}\mrightarrow{} X. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}a,b:\{X \mvdash{} \_:A\}.
(((Id\_A a b))s = (Id\_(A)s (a)s (b)s))
Date html generated:
2019_11_06-PM-00_52_24
Last ObjectModification:
2018_12_10-PM-01_54_49
Theory : cubical!sets
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