Nuprl Lemma : k-continuous-iff-all-k-1
∀[k:ℕ]. ∀[F:(ℕk ⟶ Type) ⟶ ℕk ⟶ Type]. (k-Continuous(T.F[T]) ⇐⇒ ∀i:ℕk. k-1-continuous{i:l}(k;T.F[T] i))
Proof
Definitions occuring in Statement :
k-continuous: k-Continuous(T.F[T]),
k-1-continuous: k-1-continuous{i:l}(k;T.F[T]),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
k-continuous: k-Continuous(T.F[T]),
k-1-continuous: k-1-continuous{i:l}(k;T.F[T]),
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
k-subtype: A ⊆ B,
guard: {T},
k-intersection: ⋂n. X[n]
Lemmas referenced :
int_seg_wf,
k-continuous_wf,
all_wf,
k-1-continuous_wf,
nat_wf,
k-subtype_wf,
decidable__le,
false_wf,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_wf,
subtype_rel_self,
subtype_rel_transitivity,
k-intersection_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
cumulativity,
universeEquality,
because_Cache,
instantiate,
productElimination,
independent_pairEquality,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
dependent_set_memberEquality,
addEquality,
unionElimination,
voidElimination,
independent_functionElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
voidEquality,
intEquality,
minusEquality,
isectEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[F:(\mBbbN{}k {}\mrightarrow{} Type) {}\mrightarrow{} \mBbbN{}k {}\mrightarrow{} Type].
(k-Continuous(T.F[T]) \mLeftarrow{}{}\mRightarrow{} \mforall{}i:\mBbbN{}k. k-1-continuous\{i:l\}(k;T.F[T] i))
Date html generated:
2018_05_21-PM-00_09_40
Last ObjectModification:
2017_10_18-PM-02_35_51
Theory : co-recursion
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