Nuprl Lemma : cube-context-adjoin_wf
∀[Gamma:CubicalSet]. ∀[A:{Gamma ⊢ _}]. (Gamma.A ∈ CubicalSet)
Proof
Definitions occuring in Statement :
cube-context-adjoin: 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,
cube-context-adjoin: X.A,
cubical-set: CubicalSet,
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
compose: f o g,
squash: ↓T,
prop: ℙ,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cube-set-restriction_wf,
cubical-type-ap-morph_wf,
cubical-type-at_wf,
name-morph_wf,
equal_wf,
squash_wf,
true_wf,
I-cube_wf,
name-comp_wf,
cubical-type-ap-morph-comp,
pi1_wf_top,
pi2_wf,
subtype_rel-equal,
cube-set-restriction-comp,
iff_weakening_equal,
all_wf,
compose_wf,
equal-wf-T-base,
list_wf,
coordinate_name_wf,
id-morph_wf,
cubical-type_wf,
cubical-set_wf,
cube-set-restriction-id,
cubical-type-ap-morph-id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
dependent_set_memberEquality,
dependent_pairEquality,
because_Cache,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
hypothesis,
functionEquality,
applyEquality,
functionExtensionality,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productEquality,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
instantiate,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
independent_pairFormation,
axiomEquality
Latex:
\mforall{}[Gamma:CubicalSet]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. (Gamma.A \mmember{} CubicalSet)
Date html generated:
2017_10_05-AM-10_13_17
Last ObjectModification:
2017_07_28-AM-11_18_49
Theory : cubical!sets
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