Nuprl Lemma : cubical-eta
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}]. ((λapp((w)p; q)) = w ∈ {X ⊢ _:ΠA B})
Proof
Definitions occuring in Statement :
cubical-app: app(w; u),
cubical-lambda: (λb),
cubical-pi: ΠA B,
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:AF},
cubical-type: {X ⊢ _},
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cc-snd: q,
cc-fst: p,
csm-ap-term: (t)s,
cubical-app: app(w; u),
cubical-lambda: (λb),
cubical-pi: ΠA B,
pi1: fst(t),
csm-ap: (s)x,
cubical-term: {X ⊢ _:AF},
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
squash: ↓T,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
true: True,
top: Top,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_apply: x[s],
cc-adjoin-cube: (v;u),
pi2: snd(t)
Lemmas referenced :
cubical-term_wf,
cubical-pi_wf,
cubical-type_wf,
cube-context-adjoin_wf,
cubical-set_wf,
I-cube_wf,
list_wf,
coordinate_name_wf,
cubical-type-at_wf,
cube-set-restriction_wf,
name-morph_wf,
all_wf,
equal_wf,
cc-adjoin-cube_wf,
cubical-type-ap-morph_wf,
name-comp_wf,
subtype_rel-equal,
cube-set-restriction-comp,
cc-adjoin-cube-restriction,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
id-morph_wf,
cube-set-restriction-id,
name-comp-id-right,
cubical-term-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
equalitySymmetry,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
functionExtensionality,
setElimination,
rename,
applyEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
lambdaEquality,
dependent_functionElimination,
independent_isectElimination,
natural_numberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
universeEquality,
instantiate,
productElimination,
independent_functionElimination,
hyp_replacement
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:\mPi{}A B\}]. ((\mlambda{}app((w)p; q)) = w)
Date html generated:
2018_05_23-PM-06_32_06
Last ObjectModification:
2018_05_20-PM-04_21_23
Theory : cubical!sets
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