Nuprl Lemma : discrete-family_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢]. X.discr(A) ⊢ discrete-family(A;a.B[a])
Proof
Definitions occuring in Statement :
discrete-family: discrete-family(A;a.B[a]),
discrete-cubical-type: discr(T),
cube-context-adjoin: X.A,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-type: {X ⊢ _},
discrete-family: discrete-family(A;a.B[a]),
discrete-cubical-type: discr(T),
cube-context-adjoin: X.A,
all: ∀x:A. B[x],
so_apply: x[s],
pi2: snd(t),
so_lambda: λ2x.t[x],
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
I_cube_pair_redex_lemma,
cubical_type_at_pair_lemma,
cube_set_restriction_pair_lemma,
cubical_type_ap_morph_pair_lemma,
I_cube_wf,
fset_wf,
nat_wf,
istype-universe,
names-hom_wf,
cube-set-restriction_wf,
pi1_wf_top,
pi2_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
discrete-cubical-type_wf,
nh-id_wf,
subtype_rel-equal,
equal_wf,
squash_wf,
true_wf,
cube-set-restriction-id,
subtype_rel_self,
iff_weakening_equal,
nh-comp_wf,
cube-set-restriction-comp,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
dependent_set_memberEquality_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
Error :memTop,
hypothesis,
dependent_pairEquality_alt,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
productElimination,
productIsType,
universeIsType,
isectElimination,
because_Cache,
functionIsType,
instantiate,
independent_pairEquality,
cumulativity,
lambdaFormation_alt,
independent_pairFormation,
equalityIstype,
independent_isectElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
inhabitedIsType
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[X:j\mvdash{}]. X.discr(A) \mvdash{} discrete-family(A;a.B[a])
Date html generated:
2020_05_20-PM-03_38_11
Last ObjectModification:
2020_04_06-PM-07_06_33
Theory : cubical!type!theory
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