Nuprl Lemma : cubical-snd_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[p:{X ⊢ _:Σ A B}]. (p.2 ∈ {X ⊢ _:(B)[p.1]})
Proof
Definitions occuring in Statement :
cubical-snd: p.2,
cubical-fst: p.1,
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-snd: p.2,
cubical-term: {X ⊢ _:AF},
cubical-sigma: Σ A B,
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
cubical-type: {X ⊢ _},
cubical-type-at: A(a),
csm-id-adjoin: [u],
csm-ap-type: (AF)s,
csm-adjoin: (s;u),
csm-ap: (s)x,
cc-adjoin-cube: (v;u),
cubical-fst: p.1,
csm-id: 1(X),
top: Top,
type-cat: TypeCat,
cat-id: cat-id(C),
pi2: snd(t),
and: P ∧ Q,
cubical-type-ap-morph: (u a f),
identity-trans: identity-trans(C;D;F),
cube-context-adjoin: X.A,
I-cube: X(I),
functor-ob: ob(F),
squash: ↓T,
guard: {T},
true: True,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
cubical-fst_wf,
cubical-term_wf,
cubical-sigma_wf,
cubical-type_wf,
cube-context-adjoin_wf,
cubical-set_wf,
cubical-type-at_wf,
cc-adjoin-cube_wf,
pi2_wf,
equal_wf,
I-cube_wf,
list_wf,
coordinate_name_wf,
ident_trans_ap_lemma,
subtype_rel_self,
pi1_wf_top,
name-morph_wf,
ap_mk_nat_trans_lemma,
cat_id_tuple_lemma,
ob_pair_lemma,
cube-set-restriction_wf,
squash_wf,
true_wf,
iff_weakening_equal,
subtype_rel-equal,
csm-ap_wf,
csm-id-adjoin_wf,
subtype_rel_wf,
subtype_rel_weakening,
ext-eq_weakening,
csm-ap-type_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
because_Cache,
dependent_set_memberEquality,
setElimination,
rename,
lambdaEquality,
applyEquality,
productEquality,
dependent_functionElimination,
lambdaFormation,
independent_functionElimination,
productElimination,
voidElimination,
voidEquality,
independent_pairEquality,
promote_hyp,
dependent_pairEquality,
applyLambdaEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination,
hyp_replacement,
functionEquality
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[p:\{X \mvdash{} \_:\mSigma{} A B\}]. (p.2 \mmember{} \{X \mvdash{} \_:(B)[p.1]\})
Date html generated:
2018_05_23-PM-06_31_38
Last ObjectModification:
2018_05_20-PM-04_20_00
Theory : cubical!sets
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