Nuprl Lemma : csm-cubical-isect-family
∀X,Delta:⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:Delta ⟶ X. ∀I:fset(ℕ). ∀a:Delta(I).
(cubical-isect-family(X;A;B;I;(s)a) = cubical-isect-family(Delta;(A)s;(B)(s o p;q);I;a) ∈ Type)
Proof
Definitions occuring in Statement :
cubical-isect-family: cubical-isect-family(X;A;B;I;a),
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,
csm-ap: (s)x,
cube_set_map: A ⟶ B,
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
all: ∀x:A. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
cubical-isect-family: cubical-isect-family(X;A;B;I;a),
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cube-set-restriction: f(s),
pi2: snd(t),
cube-context-adjoin: X.A,
cc-adjoin-cube: (v;u),
pi1: fst(t)
Lemmas referenced :
csm-comp_wf,
cube-context-adjoin_wf,
csm-ap-type_wf,
cc-fst_wf,
cc-snd_wf,
csm-ap-type-at,
csm-adjoin-ap,
csm_comp_fst_adjoin_cube_lemma,
cc_snd_adjoin_cube_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
names-hom_wf,
cubical-type-at_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-type_wf,
cubical_set_wf,
csm-ap-restriction,
cc-adjoin-cube_wf,
istype-cubical-type-at,
csm-ap_wf,
cube-set-restriction_wf,
subtype_rel_self,
iff_weakening_equal,
cube_set_map_wf,
cc-adjoin-cube-restriction,
cubical-type-ap-morph_wf,
subtype_rel-equal,
csm-cubical-type-ap-morph,
cube-set-restriction-comp,
nh-comp_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setEquality,
functionEquality,
because_Cache,
sqequalRule,
Error :memTop,
dependent_functionElimination,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
isectEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
inhabitedIsType,
equalityIstype,
hyp_replacement,
isect_memberEquality_alt
Latex:
\mforall{}X,Delta:\mvdash{}. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}B:\{X.A \mvdash{} \_\}. \mforall{}s:Delta {}\mrightarrow{} X. \mforall{}I:fset(\mBbbN{}). \mforall{}a:Delta(I).
(cubical-isect-family(X;A;B;I;(s)a) = cubical-isect-family(Delta;(A)s;(B)(s o p;q);I;a))
Date html generated:
2020_05_21-AM-10_47_29
Last ObjectModification:
2020_05_02-PM-10_42_01
Theory : cubical!type!theory
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