Nuprl Lemma : formal-cube-singleton1
∀x:ℕ. (λA,f. (f x) ∈ nat-trans(op-cat(CubeCat);TypeCat';formal-cube({x});𝕀))
Proof
Definitions occuring in Statement :
formal-cube: formal-cube(I),
interval-presheaf: 𝕀,
cube-cat: CubeCat,
type-cat: TypeCat,
op-cat: op-cat(C),
nat-trans: nat-trans(C;D;F;G),
fset-singleton: {x},
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
lambda: λx.A[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
ps_context: __⊢,
cubical_set: CubicalSet,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
cat-functor: Functor(C1;C2),
interval-presheaf: 𝕀,
formal-cube: formal-cube(I),
type-cat: TypeCat,
top: Top,
names-hom: I ⟶ J,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
and: P ∧ Q,
names: names(I),
prop: ℙ,
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
DeMorgan-algebra: DeMorganAlgebra,
guard: {T},
compose: f o g,
dM-lift: dM-lift(I;J;f),
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
dM: dM(I)
Lemmas referenced :
interval-presheaf_wf,
small_cubical_set_subtype,
is-nat-trans,
op-cat_wf,
cube-cat_wf,
type-cat_wf,
formal-cube_wf,
fset-singleton_wf,
nat_wf,
subtype_rel_self,
cat-functor_wf,
cat_arrow_triple_lemma,
istype-void,
ob_pair_lemma,
member-fset-singleton,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
fset-member_wf,
names_wf,
lattice-point_wf,
dM_wf,
fset_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
cat-ob_wf,
cat_comp_tuple_lemma,
arrow_pair_lemma,
cat_ob_pair_lemma,
istype-nat,
dM-lift_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
hypothesis,
applyEquality,
thin,
sqequalHypSubstitution,
sqequalRule,
instantiate,
isectElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
intEquality,
independent_isectElimination,
because_Cache,
natural_numberEquality,
productElimination,
dependent_set_memberEquality_alt,
universeIsType,
functionIsType,
productEquality,
cumulativity,
inhabitedIsType,
functionExtensionality,
rename,
functionEquality
Latex:
\mforall{}x:\mBbbN{}. (\mlambda{}A,f. (f x) \mmember{} nat-trans(op-cat(CubeCat);TypeCat';formal-cube(\{x\});\mBbbI{}))
Date html generated:
2019_11_04-PM-05_32_14
Last ObjectModification:
2018_12_13-AM-10_02_55
Theory : cubical!type!theory
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