Nuprl Lemma : discrete-fun-invariant
∀[A,B:Type]. ∀[f:{() ⊢ _:(discr(A) ⟶ discr(B))}]. ∀[I:fset(ℕ)]. ∀[a:()(I)].
((f I a) = (f {} ⋅) ∈ (discr(A) ⟶ discr(B))(a))
Proof
Definitions occuring in Statement :
discrete-cubical-type: discr(T),
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
cubical-type-at: A(a),
trivial-cube-set: (),
I_cube: A(I),
empty-fset: {},
fset: fset(T),
nat: ℕ,
it: ⋅,
uall: ∀[x:A]. B[x],
apply: f a,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
false: False,
names: names(I),
names-hom: I ⟶ J,
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
top: Top,
all: ∀x:A. B[x],
cubical-fun: (A ⟶ B),
discrete-cubical-type: discr(T),
prop: ℙ,
trivial-cube-set: (),
pi1: fst(t),
functor-ob: ob(F),
I_cube: A(I),
unit: Unit,
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
nh-comp_wf,
equal_wf,
all_wf,
member-empty-fset,
names_wf,
names-hom_wf,
cubical_type_ap_morph_pair_lemma,
cubical_type_at_pair_lemma,
cubical-type-at_wf,
equal-wf-base,
subtype_rel_self,
it_wf,
empty-fset_wf,
discrete-cubical-type_wf,
cubical-fun_wf,
cubical-term_wf,
nat_wf,
fset_wf,
trivial-cube-set_wf,
I_cube_wf,
discrete-fun-at
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
lambdaFormation,
functionExtensionality,
dependent_set_memberEquality,
rename,
setElimination,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
lambdaEquality,
because_Cache,
baseClosed,
intEquality,
sqequalRule,
applyEquality,
universeEquality,
cumulativity,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:\{() \mvdash{} \_:(discr(A) {}\mrightarrow{} discr(B))\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[a:()(I)]. ((f I a) = (f \{\} \mcdot{}))
Date html generated:
2017_02_21-AM-10_43_50
Last ObjectModification:
2017_02_13-PM-03_51_29
Theory : cubical!type!theory
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