Nuprl Lemma : list-continuity
∀[X:ℕ ⟶ Type]. ((⋂n:ℕ. (X[n] List)) ⊆r ((⋂n:ℕ. X[n]) List))
Proof
Definitions occuring in Statement :
list: T List,
nat: ℕ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s],
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
ge: i ≥ j ,
le: A ≤ B,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
less_than': less_than'(a;b),
not: ¬A,
or: P ∨ Q,
cons: [a / b],
top: Top,
exists: ∃x:A. B[x],
subtract: n - m,
uiff: uiff(P;Q),
true: True,
nat_plus: ℕ+,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
istype-nat,
list_wf,
istype-universe,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
istype-false,
istype-le,
length_wf,
subtract-1-ge-0,
nil_wf,
nat_wf,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
istype-void,
le_weakening2,
non_neg_length,
length_wf_nat,
istype-sqequal,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
subtract_wf,
le_reflexive,
one-mul,
add-mul-special,
add-associates,
two-mul,
mul-distributes-right,
zero-mul,
minus-zero,
add-swap,
omega-shadow,
decidable__lt,
decidable__le,
not-le-2,
minus-minus,
less-iff-le,
not-lt-2,
reduce_tl_nil_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
cons_wf,
hd_wf,
not-ge-2,
le-add-cancel2,
tl_wf,
le_wf,
squash_wf,
true_wf,
istype-int,
length_tl,
subtype_rel_self,
iff_weakening_equal,
le-add-cancel-alt
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaEquality_alt,
Error :isectIsType,
extract_by_obid,
hypothesis,
Error :universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
axiomEquality,
Error :functionIsType,
instantiate,
universeEquality,
Error :lambdaFormation_alt,
setElimination,
rename,
intWeakElimination,
independent_pairFormation,
productElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :dependent_set_memberEquality_alt,
because_Cache,
isectEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
Error :isect_memberEquality_alt,
Error :equalityIstype,
Error :dependent_pairFormation_alt,
addEquality,
minusEquality,
multiplyEquality,
imageMemberEquality,
baseClosed,
closedConclusion
Latex:
\mforall{}[X:\mBbbN{} {}\mrightarrow{} Type]. ((\mcap{}n:\mBbbN{}. (X[n] List)) \msubseteq{}r ((\mcap{}n:\mBbbN{}. X[n]) List))
Date html generated:
2019_06_20-PM-00_44_11
Last ObjectModification:
2019_02_21-PM-03_09_01
Theory : list_0
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