Nuprl Lemma : csm-adjoin-fst-snd
∀[Gamma,Delta:CubicalSet]. ∀[A:{Gamma ⊢ _}]. ((p;q) = 1(Gamma.A) ∈ Gamma.A ⟶ Gamma.A)
Proof
Definitions occuring in Statement :
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
cubical-type: {X ⊢ _},
csm-id: 1(X),
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
cubical-type: {X ⊢ _},
cc-snd: q,
cc-fst: p,
csm-adjoin: (s;u),
cube-context-adjoin: X.A,
csm-id: 1(X),
I-cube: X(I),
all: ∀x:A. B[x],
top: Top,
csm-ap: (s)x,
functor-ob: ob(F),
type-cat: TypeCat,
identity-trans: identity-trans(C;D;F),
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
pi1: fst(t),
pi2: snd(t),
cubical-type-at: A(a)
Lemmas referenced :
cubical-type_wf,
cubical-set_wf,
cube-context-adjoin_wf,
csm-id_wf,
csm-adjoin_wf,
cc-fst_wf,
cc-snd_wf,
cube-set-map-subtype,
csm-equal,
ob_pair_lemma,
cat_id_tuple_lemma,
ap_mk_nat_trans_lemma,
I-cube_wf,
cubical-type-at_wf,
list_wf,
coordinate_name_wf,
name-morph_wf,
cube-set-restriction_wf,
all_wf,
equal_wf,
id-morph_wf,
subtype_rel-equal,
squash_wf,
true_wf,
cube-set-restriction-id,
iff_weakening_equal,
name-comp_wf,
cube-set-restriction-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
equalitySymmetry,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
applyEquality,
independent_isectElimination,
setElimination,
rename,
productElimination,
dependent_functionElimination,
voidElimination,
voidEquality,
functionExtensionality,
productEquality,
dependent_set_memberEquality,
dependent_pairEquality,
functionEquality,
lambdaEquality,
instantiate,
imageElimination,
equalityTransitivity,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[Gamma,Delta:CubicalSet]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. ((p;q) = 1(Gamma.A))
Date html generated:
2017_10_05-AM-10_13_43
Last ObjectModification:
2017_07_28-AM-11_19_04
Theory : cubical!sets
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