Nuprl Lemma : cubical-isect-elim_wf
∀[X:⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[t:{X ⊢ _:⋂A B}]. (cubical-isect-elim(t) ∈ {X.A ⊢ _:B})
Proof
Definitions occuring in Statement :
cubical-isect-elim: cubical-isect-elim(t),
cubical-isect: ⋂A B,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-term: {X ⊢ _:A},
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
cubical-isect-elim: cubical-isect-elim(t),
cube-context-adjoin: X.A,
top: Top,
pi1: fst(t),
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
cubical-isect: ⋂A B,
cubical-isect-family: cubical-isect-family(X;A;B;I;a),
uimplies: b supposing a,
squash: ↓T,
true: True,
cc-adjoin-cube: (v;u),
pi2: snd(t),
cube-set-restriction: f(s),
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
all_wf,
fset_wf,
nat_wf,
names-hom_wf,
I_cube_wf,
cube-context-adjoin_wf,
equal_wf,
cubical-type-at_wf,
cube-set-restriction_wf,
cubical-type-ap-morph_wf,
cubical-term_wf,
cubical-isect_wf,
cubical-type_wf,
cubical_set_wf,
I_cube_pair_redex_lemma,
cubical_type_at_pair_lemma,
nh-id_wf,
cc-adjoin-cube_wf,
subtype_rel-equal,
cube-set-restriction-id,
squash_wf,
true_wf,
cube_set_restriction_pair_lemma,
cc-adjoin-cube-restriction,
subtype_rel_self,
iff_weakening_equal,
cubical_type_ap_morph_pair_lemma,
nh-id-left,
nh-id-right,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
dependent_set_memberEquality,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
sqequalRule,
lambdaEquality,
hypothesisEquality,
applyEquality,
setElimination,
rename,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
isectEquality,
independent_isectElimination,
imageElimination,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
productEquality,
applyLambdaEquality,
universeEquality,
independent_pairFormation
Latex:
\mforall{}[X:\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[t:\{X \mvdash{} \_:\mcap{}A B\}]. (cubical-isect-elim(t) \mmember{} \{X.A \mvdash{} \_:B\})
Date html generated:
2018_05_23-PM-06_27_51
Last ObjectModification:
2018_05_20-PM-09_30_16
Theory : cubical!type!theory
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