Nuprl Lemma : csm-cubical-pi-family
∀X,Delta:CubicalSet. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:Delta ⟶ X. ∀I:Cname List. ∀a:Delta(I).
(cubical-pi-family(X;A;B;I;(s)a) = cubical-pi-family(Delta;(A)s;(B)(s o p;q);I;a) ∈ Type)
Proof
Definitions occuring in Statement :
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
csm-ap: (s)x,
I-cube: X(I),
csm-comp: G o F,
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
coordinate_name: Cname,
list: T List,
all: ∀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-set: CubicalSet,
cubical-type: {X ⊢ _},
cube-set-map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
I-cube: X(I),
top: Top,
functor-arrow: arrow(F),
functor-ob: ob(F),
type-cat: TypeCat,
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
name-cat: NameCat,
cat-ob: cat-ob(C),
pi1: fst(t),
pi2: snd(t),
compose: f o g,
cube-context-adjoin: X.A,
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
cc-snd: q,
cc-fst: p,
csm-ap-type: (AF)s,
csm-comp: G o F,
csm-adjoin: (s;u),
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
csm-ap: (s)x,
cc-adjoin-cube: (v;u),
trans-comp: t1 o t2,
so_lambda: λ2x.t[x],
so_apply: x[s],
cube-set-restriction: f(s),
and: P ∧ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cand: A c∧ B
Lemmas referenced :
I-cube_wf,
list_wf,
coordinate_name_wf,
cube-set-map_wf,
cubical-type_wf,
cube-context-adjoin_wf,
ob_pair_lemma,
istype-void,
ap_mk_nat_trans_lemma,
cat_comp_tuple_lemma,
name-morph_wf,
equal_wf,
subtype_rel-equal,
squash_wf,
true_wf,
istype-universe,
subtype_rel_dep_function,
subtype_rel_self,
iff_weakening_equal,
name-comp_wf,
subtype_rel_weakening,
ext-eq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
inhabitedIsType,
setElimination,
rename,
productElimination,
sqequalRule,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
setEquality,
functionEquality,
applyEquality,
applyLambdaEquality,
dependent_pairEquality_alt,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalitySymmetry,
hyp_replacement,
dependent_set_memberEquality_alt,
equalityIsType1,
equalityTransitivity,
independent_isectElimination,
independent_pairFormation,
productIsType,
instantiate,
universeEquality,
independent_functionElimination,
independent_pairEquality
Latex:
\mforall{}X,Delta:CubicalSet. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}B:\{X.A \mvdash{} \_\}. \mforall{}s:Delta {}\mrightarrow{} X. \mforall{}I:Cname List. \mforall{}a:Delta(I).
(cubical-pi-family(X;A;B;I;(s)a) = cubical-pi-family(Delta;(A)s;(B)(s o p;q);I;a))
Date html generated:
2019_11_05-PM-00_26_29
Last ObjectModification:
2018_11_10-PM-02_44_54
Theory : cubical!sets
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