Nuprl Lemma : csm-adjoin_wf
∀[Gamma,Delta:CubicalSet]. ∀[A:{Gamma ⊢ _}]. ∀[sigma:Delta ⟶ Gamma]. ∀[u:{Delta ⊢ _:(A)sigma}].
((sigma;u) ∈ Delta ⟶ Gamma.A)
Proof
Definitions occuring in Statement :
csm-adjoin: (s;u),
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:AF},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
csm-adjoin: (s;u),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
cc-adjoin-cube: (v;u),
cubical-term: {X ⊢ _:AF},
cubical-type-at: A(a),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True,
cubical-term-at: u(a)
Lemmas referenced :
cube-set-map-is,
cube-context-adjoin_wf,
I-cube_wf,
list_wf,
coordinate_name_wf,
name-morph_wf,
all_wf,
equal_wf,
cube-set-restriction_wf,
cubical-term_wf,
csm-ap-type_wf,
cube-set-map_wf,
cubical-type_wf,
cubical-set_wf,
cc-adjoin-cube_wf,
csm-ap_wf,
subtype_rel-equal,
cubical-type-at_wf,
csm-type-at,
cc-adjoin-cube-restriction,
squash_wf,
true_wf,
csm-ap-restriction,
iff_weakening_equal,
cubical-type-ap-morph_wf,
cubical-term-at_wf,
cubical-term-at-morph,
csm-cubical-type-ap-morph
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
lambdaEquality,
lambdaFormation,
because_Cache,
functionEquality,
applyEquality,
functionExtensionality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
dependent_functionElimination,
setElimination,
rename,
independent_isectElimination,
voidElimination,
voidEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
natural_numberEquality,
instantiate
Latex:
\mforall{}[Gamma,Delta:CubicalSet]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[sigma:Delta {}\mrightarrow{} Gamma]. \mforall{}[u:\{Delta \mvdash{} \_:(A)sigma\}].
((sigma;u) \mmember{} Delta {}\mrightarrow{} Gamma.A)
Date html generated:
2017_10_05-AM-10_13_31
Last ObjectModification:
2017_07_28-AM-11_18_56
Theory : cubical!sets
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