Nuprl Lemma : discrete-pair-inv-property
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢]. ∀[b:{X ⊢ _:discr(a:A × B[a])}].
(discrete-pair(discrete-pair-inv(X;b)) = b ∈ {X ⊢ _:discr(a:A × B[a])})
Proof
Definitions occuring in Statement :
discrete-pair-inv: discrete-pair-inv(X;b),
discrete-pair: discrete-pair(p),
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],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
discrete-pair-inv: discrete-pair-inv(X;b),
so_apply: x[s],
uimplies: b supposing a,
discrete-pair: discrete-pair(p),
discrete-cubical-type: discr(T),
all: ∀x:A. B[x],
cubical-snd: p.2,
cubical-fst: p.1,
cubical-term-at: u(a),
cubical-pair: cubical-pair(u;v),
pi1: fst(t),
pi2: snd(t),
implies: P ⇒ Q
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,
cubical-term-at_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productEquality,
applyEquality,
equalityTransitivity,
independent_isectElimination,
universeIsType,
instantiate,
cumulativity,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
universeEquality,
dependent_functionElimination,
Error :memTop,
lambdaFormation_alt,
productElimination,
dependent_pairEquality_alt,
equalityIstype,
independent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[X:j\mvdash{}]. \mforall{}[b:\{X \mvdash{} \_:discr(a:A \mtimes{} B[a])\}].
(discrete-pair(discrete-pair-inv(X;b)) = b)
Date html generated:
2020_05_20-PM-03_40_58
Last ObjectModification:
2020_04_06-PM-07_12_28
Theory : cubical!type!theory
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