Nuprl Lemma : discrete-fun-app-invariant
∀[A,B:Type]. ∀[f:{() ⊢ _:(discr(A) ⟶ discr(B))}]. ∀[I:fset(ℕ)]. ∀[a:()(I)]. ∀[t:A].
(app(f; discr(t))(a) = app(f; discr(t))(⋅) ∈ B)
Proof
Definitions occuring in Statement :
discrete-cubical-term: discr(t),
discrete-cubical-type: discr(T),
cubical-app: app(w; u),
cubical-fun: (A ⟶ B),
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
trivial-cube-set: (),
I_cube: A(I),
empty-fset: {},
fset: fset(T),
nat: ℕ,
it: ⋅,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
cubical-term-at: u(a),
uall: ∀[x:A]. B[x],
member: t ∈ T,
discrete-cubical-term: discr(t),
cubical-app: app(w; u),
discrete-cubical-type: discr(T),
cubical-fun: (A ⟶ B),
all: ∀x:A. B[x],
top: Top,
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
squash: ↓T,
cubical-term: {X ⊢ _:A},
subtype_rel: A ⊆r B,
unit: Unit,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
trivial-cube-set: (),
prop: ℙ,
implies: P ⇒ Q,
names-hom: I ⟶ J,
names: names(I),
false: False,
true: True,
cubical-type-at: A(a),
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
discrete-fun-invariant,
I_cube_wf,
trivial-cube-set_wf,
fset_wf,
nat_wf,
cubical-term_wf,
cubical-fun_wf,
discrete-cubical-type_wf,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
empty-fset_wf,
it_wf,
subtype_rel_self,
equal-wf-base,
cubical-type-at_wf,
equal_wf,
dM0_wf,
names_wf,
member-empty-fset,
squash_wf,
true_wf,
nh-id_wf,
subtype_rel_dep_function,
names-hom_wf,
cube-set-restriction_wf,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
applyLambdaEquality,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
functionExtensionality,
intEquality,
because_Cache,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
lambdaEquality,
hyp_replacement,
natural_numberEquality,
functionEquality,
independent_isectElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:\{() \mvdash{} \_:(discr(A) {}\mrightarrow{} discr(B))\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[a:()(I)]. \mforall{}[t:A].
(app(f; discr(t))(a) = app(f; discr(t))(\mcdot{}))
Date html generated:
2017_10_05-AM-02_12_20
Last ObjectModification:
2017_03_02-PM-11_21_26
Theory : cubical!type!theory
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