Nuprl Lemma : cubical-lambda_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[b:{X.A ⊢ _:B}]. ((λb) ∈ {X ⊢ _:ΠA B})
Proof
Definitions occuring in Statement :
cubical-lambda: (λb),
cubical-pi: ΠA B,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:AF},
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 ⊢ _:AF},
so_apply: x[s],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
prop: ℙ,
squash: ↓T,
top: Top,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
cubical-type-at: A(a),
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
pi1: fst(t),
cubical-pi: ΠA B,
cubical-lambda: (λb),
pi2: snd(t),
cubical-type-ap-morph: (u a f),
cubical-type: {X ⊢ _},
cc-adjoin-cube: (v;u),
cube-set-restriction: f(s),
cube-context-adjoin: X.A
Lemmas referenced :
cubical-term_wf,
cube-context-adjoin_wf,
cubical-type_wf,
cubical-set_wf,
I-cube_wf,
iff_weakening_equal,
cube-set-restriction-comp,
name-comp_wf,
true_wf,
squash_wf,
cc-adjoin-cube-restriction,
subtype_rel-equal,
cubical-type-ap-morph_wf,
equal_wf,
all_wf,
coordinate_name_wf,
list_wf,
name-morph_wf,
cubical-type-at_wf,
cube-set-restriction_wf,
cc-adjoin-cube_wf,
subtype_rel_self,
istype-universe,
cubical-pi_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
independent_functionElimination,
productElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
imageElimination,
instantiate,
voidEquality,
voidElimination,
isect_memberEquality,
independent_isectElimination,
functionExtensionality,
lambdaFormation,
dependent_functionElimination,
because_Cache,
rename,
setElimination,
applyEquality,
dependent_set_memberEquality,
lambdaEquality,
hyp_replacement,
lambdaFormation_alt,
functionIsType,
equalityIstype,
lambdaEquality_alt,
Error :memTop,
functionExtensionality_alt
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[b:\{X.A \mvdash{} \_:B\}]. ((\mlambda{}b) \mmember{} \{X \mvdash{} \_:\mPi{}A B\})
Date html generated:
2020_05_21-AM-10_50_13
Last ObjectModification:
2020_01_01-PM-02_47_27
Theory : cubical!sets
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