Nuprl Lemma : csm-cubical-sigma
∀X,Delta:CubicalSet. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:Delta ⟶ X. ((Σ A B)s = Σ (A)s (B)(s o p;q) ∈ {Delta ⊢ _})
Proof
Definitions occuring in Statement :
cubical-sigma: Σ A B,
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
csm-comp: G o F,
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cubical-type: {X ⊢ _},
cube-set-map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
csm-ap-type: (AF)s,
cubical-sigma: Σ A B,
cubical-type-at: A(a),
pi1: fst(t),
and: P ∧ Q,
top: Top,
cubical-type-ap-morph: (u a f),
pi2: snd(t),
subtype_rel: A ⊆r B,
cc-adjoin-cube: (v;u),
cc-snd: q,
csm-ap: (s)x,
cc-fst: p,
csm-comp: G o F,
type-cat: TypeCat,
cat-comp: cat-comp(C),
compose: f o g,
cube-context-adjoin: X.A,
I-cube: X(I),
functor-ob: ob(F),
cat-arrow: cat-arrow(C),
name-cat: NameCat,
cat-ob: cat-ob(C),
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
csm-adjoin: (s;u),
trans-comp: t1 o t2,
so_lambda: λ2x.t[x],
so_apply: x[s],
functor-arrow: arrow(F),
cube-set-restriction: f(s)
Lemmas referenced :
cubical-type-equal,
csm-ap-type_wf,
cubical-sigma_wf,
cube-set-map_wf,
cubical-type_wf,
cube-context-adjoin_wf,
cubical-set_wf,
csm-ap_wf,
csm-adjoin-ap,
istype-void,
I-cube_wf,
list_wf,
coordinate_name_wf,
cc-adjoin-cube_wf,
subtype_rel_self,
cubical-type-at_wf,
name-morph_wf,
cube-set-restriction_wf,
trans_comp_ap_lemma,
ob_pair_lemma,
cat_comp_tuple_lemma,
subtype_rel-equal,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-ap-restriction,
iff_weakening_equal,
ap_mk_nat_trans_lemma,
cc-adjoin-cube-restriction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
universeIsType,
inhabitedIsType,
setElimination,
rename,
productElimination,
sqequalRule,
dependent_pairEquality_alt,
lambdaEquality_alt,
productEquality,
applyEquality,
isect_memberEquality_alt,
voidElimination,
functionExtensionality,
because_Cache,
dependent_functionElimination,
functionIsType,
instantiate,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
applyLambdaEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIsType1
Latex:
\mforall{}X,Delta:CubicalSet. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}B:\{X.A \mvdash{} \_\}. \mforall{}s:Delta {}\mrightarrow{} X. ((\mSigma{} A B)s = \mSigma{} (A)s (B)(s o p;q))
Date html generated:
2019_11_05-PM-00_26_42
Last ObjectModification:
2018_11_10-PM-03_06_05
Theory : cubical!sets
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