Nuprl Lemma : discrete-pair-inv_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢]. ∀[b:{X ⊢ _:discr(a:A × B[a])}].
(discrete-pair-inv(X;b) ∈ {X ⊢ _:Σ discr(A) discrete-family(A;a.B[a])})
Proof
Definitions occuring in Statement :
discrete-pair-inv: discrete-pair-inv(X;b),
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,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-term: {X ⊢ _:A},
discrete-cubical-type: discr(T),
all: ∀x:A. B[x],
cubical-term-at: u(a),
implies: P ⇒ Q,
pi1: fst(t),
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
discrete-pair-inv: discrete-pair-inv(X;b),
discrete-family: discrete-family(A;a.B[a]),
csm-id-adjoin: [u],
csm-ap-type: (AF)s,
csm-adjoin: (s;u),
csm-ap: (s)x,
pi2: snd(t)
Lemmas referenced :
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
I_cube_wf,
fset_wf,
nat_wf,
equal_wf,
squash_wf,
true_wf,
pi1_wf_top,
istype-top,
subtype_rel_product,
top_wf,
subtype_rel_self,
iff_weakening_equal,
names-hom_wf,
istype-cubical-type-at,
cube-set-restriction_wf,
discrete-cubical-type_wf,
cubical-type-ap-morph_wf,
cubical-pair_wf,
discrete-family_wf,
cubical-term_wf,
cubical_set_wf,
istype-universe,
csm-ap-type_wf,
cube-context-adjoin_wf,
csm-id-adjoin_wf,
cubical-term-at_wf,
pi2_wf,
csm-ap-type-at,
discrete-cubical-term-at-morph,
subtype_rel-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
sqequalRule,
introduction,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
inhabitedIsType,
lambdaFormation_alt,
productElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
universeIsType,
isectElimination,
imageElimination,
because_Cache,
productIsType,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
functionIsType,
cumulativity,
productEquality,
universeEquality,
independent_pairFormation,
applyLambdaEquality
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-inv(X;b) \mmember{} \{X \mvdash{} \_:\mSigma{} discr(A) discrete-family(A;a.B[a])\})
Date html generated:
2020_05_20-PM-03_40_46
Last ObjectModification:
2020_04_07-PM-04_29_55
Theory : cubical!type!theory
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