Nuprl Lemma : discrete-function-inv-property
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢]. ∀[b:{X ⊢ _:discr(a:A ⟶ B[a])}].
(discrete-function(discrete-function-inv(X; b)) = b ∈ {X ⊢ _:discr(a:A ⟶ B[a])})
Proof
Definitions occuring in Statement :
discrete-function-inv: discrete-function-inv(X; b),
discrete-function: discrete-function(f),
discrete-cubical-type: discr(T),
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
discrete-function-inv: discrete-function-inv(X; b),
so_apply: x[s],
uimplies: b supposing a,
cubical-lambda: (λb),
discrete-function: discrete-function(f),
discrete-cubical-type: discr(T),
all: ∀x:A. B[x],
cc-adjoin-cube: (v;u),
cc-snd: q,
cubical-term-at: u(a),
pi1: fst(t),
pi2: snd(t),
cubical-term: {X ⊢ _:A},
subtype_rel: A ⊆r B,
cubical-type-at: A(a)
Lemmas referenced :
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
discrete-cubical-type_wf,
cubical-term_wf,
cubical_set_wf,
istype-universe,
cubical_type_at_pair_lemma,
subtype_rel_self,
cube-set-restriction-id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
functionEquality,
applyEquality,
equalityTransitivity,
independent_isectElimination,
universeIsType,
instantiate,
cumulativity,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
universeEquality,
dependent_functionElimination,
Error :memTop,
setElimination,
rename
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(discrete-function-inv(X; b)) = b)
Date html generated:
2020_05_20-PM-03_39_16
Last ObjectModification:
2020_04_06-PM-07_08_29
Theory : cubical!type!theory
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