Nuprl Lemma : strong-continuity2-iff-3
∀[T:Type]. ∀[F:(ℕ ⟶ T) ⟶ ℕ]. (strong-continuity2(T;F) ⇐⇒ strong-continuity3(T;F))
Proof
Definitions occuring in Statement :
strong-continuity3: strong-continuity3(T;F),
strong-continuity2: strong-continuity2(T;F),
nat: ℕ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
sq_type: SQType(T),
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
uimplies: b supposing a,
nat: ℕ,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
cand: A c∧ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
strong-continuity2: strong-continuity2(T;F),
strong-continuity3: strong-continuity3(T;F),
rev_implies: P ⇐ Q,
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
and_wf,
int_subtype_base,
le_wf,
set_subtype_base,
subtype_base_sq,
strong-continuity3_wf,
isect_wf,
exists_wf,
all_wf,
isl_wf,
assert_wf,
false_wf,
int_seg_subtype_nat,
int_seg_wf,
subtype_rel_dep_function,
unit_wf2,
equal_wf,
nat_wf,
strong-continuity2_wf,
strong-continuity2-implies-3
Rules used in proof :
applyLambdaEquality,
dependent_set_memberEquality,
intEquality,
instantiate,
universeEquality,
inlEquality,
productEquality,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
rename,
setElimination,
natural_numberEquality,
lambdaEquality,
sqequalRule,
because_Cache,
unionEquality,
dependent_functionElimination,
dependent_pairFormation,
productElimination,
functionEquality,
applyEquality,
functionExtensionality,
cumulativity,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. \mforall{}[F:(\mBbbN{} {}\mrightarrow{} T) {}\mrightarrow{} \mBbbN{}]. (strong-continuity2(T;F) \mLeftarrow{}{}\mRightarrow{} strong-continuity3(T;F))
Date html generated:
2017_09_29-PM-06_05_14
Last ObjectModification:
2017_09_04-PM-00_14_51
Theory : continuity
Home
Index