Nuprl Lemma : discrete-function-inv_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢]. ∀[b:{X ⊢ _:discr(a:A ⟶ B[a])}].
(discrete-function-inv(X; b) ∈ {X ⊢ _:Πdiscr(A) discrete-family(A;a.B[a])})
Proof
Definitions occuring in Statement :
discrete-function-inv: discrete-function-inv(X; b),
discrete-family: discrete-family(A;a.B[a]),
discrete-cubical-type: discr(T),
cubical-pi: ΠA B,
cubical-term: {X ⊢ _:A},
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,
discrete-function-inv: discrete-function-inv(X; b),
so_lambda: λ2x.t[x],
so_apply: x[s],
cubical-term: {X ⊢ _:A},
discrete-cubical-type: discr(T),
all: ∀x:A. B[x],
cc-snd: q,
cube-context-adjoin: X.A,
discrete-family: discrete-family(A;a.B[a]),
cubical-term-at: u(a),
pi2: snd(t),
pi1: fst(t),
guard: {T},
subtype_rel: A ⊆r B
Lemmas referenced :
cubical-lambda_wf,
discrete-cubical-type_wf,
discrete-family_wf,
cubical-term_wf,
cubical_set_wf,
istype-universe,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
I_cube_pair_redex_lemma,
pi1_wf_top,
I_cube_wf,
pi2_wf,
fset_wf,
nat_wf,
cube_set_restriction_pair_lemma,
names-hom_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
istype-cubical-type-at,
cube-set-restriction_wf,
cubical-type-ap-morph_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
hypothesis,
lambdaEquality_alt,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
instantiate,
cumulativity,
functionEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
universeEquality,
setElimination,
rename,
dependent_set_memberEquality_alt,
dependent_functionElimination,
Error :memTop,
productElimination,
independent_pairEquality,
productIsType,
lambdaFormation_alt,
equalityIstype
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[X:j\mvdash{}]. \mforall{}[b:\{X \mvdash{} \_:discr(a:A {}\mrightarrow{} B[a])\}].
(discrete-function-inv(X; b) \mmember{} \{X \mvdash{} \_:\mPi{}discr(A) discrete-family(A;a.B[a])\})
Date html generated:
2020_05_20-PM-03_39_04
Last ObjectModification:
2020_04_06-PM-07_07_53
Theory : cubical!type!theory
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