Nuprl Lemma : cube-_wf
∀[I:fset(ℕ)]. ∀[i:ℕ]. (cube-(I;i) ∈ formal-cube(I+i) j⟶ formal-cube(I).𝕀)
Proof
Definitions occuring in Statement :
cube-: cube-(I;i),
interval-type: 𝕀,
cube-context-adjoin: X.A,
cube_set_map: A ⟶ B,
formal-cube: formal-cube(I),
add-name: I+i,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
nat-trans: nat-trans(C;D;F;G),
psc_map: A ⟶ B,
cube_set_map: A ⟶ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
nat: ℕ,
names: names(I),
istype: istype(T),
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
names-hom: I ⟶ J,
interval-presheaf: 𝕀,
cube-: cube-(I;i),
all: ∀x:A. B[x],
cube-context-adjoin: X.A,
pi1: fst(t),
pi2: snd(t),
spreadn: spread4,
cube-cat: CubeCat,
cat-ob: cat-ob(C),
op-cat: op-cat(C),
cat-arrow: cat-arrow(C),
type-cat: TypeCat,
functor-ob: ob(F),
formal-cube: formal-cube(I),
functor-arrow: arrow(F),
compose: f o g,
guard: {T},
and: P ∧ Q,
DeMorgan-algebra: DeMorganAlgebra,
fset: fset(T),
cube-set-restriction: f(s),
I_cube: A(I)
Lemmas referenced :
nat_wf,
fset_wf,
istype-nat,
names-hom_wf,
strong-subtype-self,
istype-int,
le_wf,
strong-subtype-set3,
strong-subtype-deq-subtype,
int-deq_wf,
fset-member_wf,
trivial-member-add-name1,
f-subset-add-name,
names-subtype,
dM_wf,
lattice-point_wf,
add-name_wf,
names_wf,
subtype_rel_dep_function,
interval-type-at,
I_cube_pair_redex_lemma,
cube_set_restriction_pair_lemma,
arrow_pair_lemma,
cat_comp_tuple_lemma,
nh-comp_wf,
f-subset_weakening,
names-hom-subtype,
interval-type-ap-morph,
nh-comp-sq,
dM-lift_wf2,
DeMorgan-algebra-axioms_wf,
lattice-join_wf,
lattice-meet_wf,
equal_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure_wf,
subtype_rel_transitivity,
DeMorgan-algebra-structure-subtype,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
DeMorgan-algebra-structure_wf,
subtype_rel_set,
subtype_rel_self,
cube-set-restriction_wf,
interval-type_wf,
cube-context-adjoin_wf,
formal-cube_wf1,
I_cube_wf,
cat-arrow_wf,
cube-cat_wf,
op-cat_wf,
cat-ob_wf,
cat_arrow_triple_lemma
Rules used in proof :
thin,
isectElimination,
sqequalHypSubstitution,
universeIsType,
hypothesis,
extract_by_obid,
introduction,
cut,
dependent_set_memberEquality_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
natural_numberEquality,
intEquality,
lambdaFormation_alt,
independent_isectElimination,
because_Cache,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
dependent_pairEquality_alt,
functionExtensionality,
Error :memTop,
dependent_functionElimination,
sqequalRule,
inhabitedIsType,
isectEquality,
cumulativity,
productEquality,
instantiate,
equalityIstype,
functionIsType
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\mBbbN{}]. (cube-(I;i) \mmember{} formal-cube(I+i) j{}\mrightarrow{} formal-cube(I).\mBbbI{})
Date html generated:
2020_05_20-PM-02_38_43
Last ObjectModification:
2020_04_04-PM-01_34_57
Theory : cubical!type!theory
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