Nuprl Lemma : discrete-fun-at
∀[A,B:Type]. ∀[f:{() ⊢ _:(discr(A) ⟶ discr(B))}]. ∀[I:fset(ℕ)]. ∀[a:()(I)].
((f I a) = (λJ,h,u. (f {} ⋅ {} 1 u)) ∈ (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),
nh-id: 1,
empty-fset: {},
fset: fset(T),
nat: ℕ,
it: ⋅,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
discrete-cubical-type: discr(T),
cubical-fun: (A ⟶ B),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
all: ∀x:A. B[x],
top: Top,
cubical-term: {X ⊢ _:A},
cubical-type-at: A(a),
pi1: fst(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
trivial-cube-set: (),
cubical-type-ap-morph: (u a f),
pi2: snd(t),
subtype_rel: A ⊆r B,
I_cube: A(I),
functor-ob: ob(F),
unit: Unit,
squash: ↓T,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
names-hom: I ⟶ J,
names: names(I),
false: False,
nh-id: 1,
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
compose: f o g,
dM: dM(I),
dM-lift: dM-lift(I;J;f)
Lemmas referenced :
I_cube_wf,
fset_wf,
nat_wf,
cubical-term_wf,
trivial-cube-set_wf,
cubical-fun_wf,
discrete-cubical-type_wf,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
names-hom_wf,
all_wf,
equal_wf,
nh-comp_wf,
I_cube_pair_redex_lemma,
cube_set_restriction_pair_lemma,
subtype_rel_self,
equal-wf-base,
nh-id_wf,
squash_wf,
true_wf,
nh-id-left,
iff_weakening_equal,
empty-fset_wf,
dM0_wf,
names_wf,
it_wf,
member-empty-fset,
dM-lift-0-sq,
set_wf,
nh-id-right
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
cumulativity,
universeEquality,
promote_hyp,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
dependent_set_memberEquality,
lambdaFormation,
lambdaEquality,
applyEquality,
functionExtensionality,
intEquality,
baseClosed,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
setEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
functionEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:\{() \mvdash{} \_:(discr(A) {}\mrightarrow{} discr(B))\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[a:()(I)].
((f I a) = (\mlambda{}J,h,u. (f \{\} \mcdot{} \{\} 1 u)))
Date html generated:
2017_10_05-AM-02_12_05
Last ObjectModification:
2017_03_02-PM-11_21_53
Theory : cubical!type!theory
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