Nuprl Lemma : csm-cubical-isect
∀X,Delta:CubicalSet. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:Delta ⟶ X. ((⋂A B)s = ⋂(A)s (B)(s o p;q) ∈ {Delta ⊢ _})
Proof
Definitions occuring in Statement :
cubical-isect: ⋂A B,
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
csm-comp: G o F,
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-type: {X ⊢ _},
cc-snd: q,
csm-ap-type: (AF)s,
cc-fst: p,
csm-comp: G o F,
csm-ap: (s)x,
compose: f o g,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
cubical-isect: ⋂A B,
cubical-isect-family: cubical-isect-family(X;A;B;I;a)
Lemmas referenced :
cubical-type-equal,
csm-ap-type_wf,
cubical-isect_wf,
cube-context-adjoin_wf,
csm-adjoin_wf,
csm-comp_wf,
cc-fst_wf,
cc-snd_wf,
cube_set_map_wf,
cubical-type_wf,
cubical_set_wf,
I_cube_wf,
fset_wf,
nat_wf,
csm-cubical-isect-family,
cubical-isect-family_wf,
csm-ap_wf,
names-hom_wf,
cubical-isect-family-comp,
cube-set-restriction_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
setElimination,
rename,
productElimination,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
applyEquality,
independent_isectElimination,
dependent_pairEquality,
lambdaEquality,
dependent_functionElimination,
functionEquality,
functionExtensionality
Latex:
\mforall{}X,Delta:CubicalSet. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}B:\{X.A \mvdash{} \_\}. \mforall{}s:Delta {}\mrightarrow{} X. ((\mcap{}A B)s = \mcap{}(A)s (B)(s o p;q))
Date html generated:
2016_10_28-AM-11_16_38
Last ObjectModification:
2016_07_21-AM-11_49_14
Theory : cubical!type!theory
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