Nuprl Lemma : cubical-sigma_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. X ⊢ Σ A B
Proof
Definitions occuring in Statement :
cubical-sigma: Σ A B,
cube-context-adjoin: X.A,
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-sigma: Σ A B,
cubical-type: {X ⊢ _},
all: ∀x:A. B[x],
pi1: fst(t),
pi2: snd(t),
subtype_rel: A ⊆r B,
top: Top,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_apply: x[s],
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a)
Lemmas referenced :
cubical-type-at_wf,
cube-context-adjoin_wf,
cc-adjoin-cube_wf,
I-cube_wf,
list_wf,
coordinate_name_wf,
cubical-type-ap-morph_wf,
subtype_rel-equal,
cube-set-restriction_wf,
pi1_wf_top,
cc-adjoin-cube-restriction,
name-morph_wf,
all_wf,
equal_wf,
id-morph_wf,
squash_wf,
true_wf,
cube-set-restriction-id,
iff_weakening_equal,
name-comp_wf,
cube-set-restriction-comp,
cubical-type_wf,
cubical-set_wf,
cubical-type-ap-morph-id,
subtype_rel_self,
subtype_rel_weakening,
ext-eq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
dependent_pairEquality,
lambdaEquality,
productEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
sqequalRule,
productElimination,
because_Cache,
applyEquality,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
functionEquality,
functionExtensionality,
lambdaFormation,
independent_pairFormation,
spreadEquality,
instantiate,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
axiomEquality,
setElimination,
rename,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. X \mvdash{} \mSigma{} A B
Date html generated:
2017_10_05-AM-10_15_21
Last ObjectModification:
2017_07_28-AM-11_19_29
Theory : cubical!sets
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