Nuprl Lemma : discrete-pair-injection
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢].
∀f,g:{X ⊢ _:Σ discr(A) discrete-family(A;a.B[a])}.
f = g ∈ {X ⊢ _:Σ discr(A) discrete-family(A;a.B[a])}
supposing discrete-pair(f) = discrete-pair(g) ∈ {X ⊢ _:discr(a:A × B[a])}
Proof
Definitions occuring in Statement :
discrete-pair: discrete-pair(p),
discrete-family: discrete-family(A;a.B[a]),
discrete-cubical-type: discr(T),
cubical-sigma: Σ A B,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
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,
all: ∀x:A. B[x],
uimplies: b supposing a,
cubical-term-at: u(a),
so_lambda: λ2x.t[x],
so_apply: x[s],
cubical-sigma: Σ A B,
cc-adjoin-cube: (v;u),
discrete-family: discrete-family(A;a.B[a]),
discrete-cubical-type: discr(T),
pi2: snd(t),
subtype_rel: A ⊆r B,
discrete-pair: discrete-pair(p),
cubical-snd: p.2,
cubical-fst: p.1,
guard: {T}
Lemmas referenced :
cubical-term-at_wf,
cubical-sigma_wf,
discrete-cubical-type_wf,
discrete-family_wf,
cubical_type_at_pair_lemma,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
cubical-term_wf,
discrete-pair_wf,
cubical_set_wf,
istype-universe,
pair-eta,
subtype_rel_product,
top_wf,
istype-top
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
applyEquality,
universeIsType,
dependent_functionElimination,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
equalityIstype,
instantiate,
cumulativity,
productEquality,
because_Cache,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
functionIsTypeImplies,
functionIsType,
universeEquality,
applyLambdaEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[X:j\mvdash{}].
\mforall{}f,g:\{X \mvdash{} \_:\mSigma{} discr(A) discrete-family(A;a.B[a])\}.
f = g supposing discrete-pair(f) = discrete-pair(g)
Date html generated:
2020_05_20-PM-03_41_11
Last ObjectModification:
2020_04_06-PM-07_13_54
Theory : cubical!type!theory
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