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 :
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat-trans: nat-trans(C;D;F;G),
psc_map: A ⟶ B,
cube_set_map: A ⟶ B,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
DeMorgan-algebra: DeMorganAlgebra,
subtype_rel: A ⊆r B,
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
nat: ℕ,
names: names(I),
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),
compose: f o g,
functor-arrow: arrow(F),
fset: fset(T),
cube-set-restriction: f(s),
I_cube: A(I)
Lemmas referenced :
nat_wf,
fset_wf,
istype-nat,
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,
dM_wf,
lattice-point_wf,
names-hom_wf,
add-name_wf,
names_wf,
not-added-name,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
eq_int_wf,
interval-type-at,
I_cube_pair_redex_lemma,
cube_set_restriction_pair_lemma,
arrow_pair_lemma,
cat_comp_tuple_lemma,
nh-comp-sq,
assert_of_eq_int,
eqtt_to_assert,
dM-lift_wf2,
interval-type-ap-morph,
subtype_rel_self,
cube-set-restriction_wf,
interval-type_wf,
formal-cube_wf1,
cube-context-adjoin_wf,
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,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_set_memberEquality_alt,
isectEquality,
productEquality,
applyEquality,
voidElimination,
independent_functionElimination,
cumulativity,
instantiate,
promote_hyp,
equalityIstype,
dependent_pairFormation_alt,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
hypothesisEquality,
equalityElimination,
unionElimination,
lambdaFormation_alt,
inhabitedIsType,
because_Cache,
rename,
setElimination,
lambdaEquality_alt,
productElimination,
functionExtensionality,
Error :memTop,
dependent_functionElimination,
sqequalRule,
functionIsType
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\mBbbN{}]. (cube+(I;i) \mmember{} formal-cube(I).\mBbbI{} j{}\mrightarrow{} formal-cube(I+i))
Date html generated:
2020_05_20-PM-02_38_31
Last ObjectModification:
2020_04_04-PM-01_34_45
Theory : cubical!type!theory
Home
Index