Nuprl Lemma : cubical-pair_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[v:{X ⊢ _:(B)[u]}]. (cubical-pair(u;v) ∈ {X ⊢ _:Σ A B})
Proof
Definitions occuring in Statement :
cubical-pair: cubical-pair(u;v),
cubical-sigma: Σ A B,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:AF},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-pair: cubical-pair(u;v),
cubical-term: {X ⊢ _:AF},
cubical-sigma: Σ A B,
pi1: fst(t),
cubical-type-at: A(a),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
uimplies: b supposing a,
top: Top,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
pi2: snd(t),
cubical-term-at: u(a),
cubical-type: {X ⊢ _},
cubical-type-ap-morph: (u a f),
csm-ap-type: (AF)s,
csm-id-adjoin: [u],
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x,
type-cat: TypeCat,
cat-id: cat-id(C),
cc-adjoin-cube: (v;u),
identity-trans: identity-trans(C;D;F),
mk-nat-trans: x |→ T[x],
cube-set-restriction: f(s),
cube-context-adjoin: X.A,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
cubical-term_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
csm-id-adjoin_wf,
cubical-type_wf,
cubical-set_wf,
subtype_rel_self,
cubical-type-at_wf,
subtype_rel-equal,
cc-adjoin-cube_wf,
csm-type-at,
equal_wf,
squash_wf,
true_wf,
csm-id-adjoin-ap,
iff_weakening_equal,
I-cube_wf,
list_wf,
coordinate_name_wf,
cubical-term-at-morph,
cube-set-restriction_wf,
name-morph_wf,
ident_trans_ap_lemma,
csm-ap_wf,
subtype_rel_weakening,
ext-eq_weakening,
cubical-sigma_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality,
because_Cache,
setElimination,
rename,
dependent_set_memberEquality,
lambdaEquality,
dependent_pairEquality,
applyEquality,
dependent_functionElimination,
independent_isectElimination,
voidElimination,
voidEquality,
instantiate,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
promote_hyp,
lambdaFormation,
functionExtensionality,
functionEquality,
hyp_replacement
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[u:\{X \mvdash{} \_:A\}]. \mforall{}[v:\{X \mvdash{} \_:(B)[u]\}].
(cubical-pair(u;v) \mmember{} \{X \mvdash{} \_:\mSigma{} A B\})
Date html generated:
2018_05_23-PM-06_31_46
Last ObjectModification:
2018_05_20-PM-04_20_21
Theory : cubical!sets
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