Nuprl Lemma : strong-continuity-implies4
∀[F:(ℕ ⟶ ℕ) ⟶ ℕ]
(↓∃M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕn?)
∀f:ℕ ⟶ ℕ. ((∃n:ℕ. ((M n f) = (inl (F f)) ∈ (ℕ?))) ∧ (∀m:ℕ. ((↑isl(M m f)) ⇒ ((M m f) = (inl (F f)) ∈ (ℕ?))))))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
isl: isl(x),
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P)
Lemmas referenced :
strong-continuity-implies3,
istype-nat,
unit_wf2,
subtype_rel_function,
nat_wf,
int_seg_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
union_subtype_base,
set_subtype_base,
lelt_wf,
istype-int,
int_subtype_base,
unit_subtype_base,
istype-assert,
btrue_wf,
bfalse_wf,
isl_wf,
mu-property,
subtype_base_sq,
bool_wf,
bool_subtype_base,
mu_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
sq_stable__all,
assert_wf,
equal-wf-base-T,
sq_stable__equal,
subtype_rel_union,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
Error :dependent_pairFormation_alt,
Error :lambdaFormation_alt,
hypothesis,
dependent_functionElimination,
Error :functionIsType,
because_Cache,
sqequalRule,
Error :productIsType,
Error :equalityIstype,
Error :unionIsType,
Error :universeIsType,
applyEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
intEquality,
Error :lambdaEquality_alt,
closedConclusion,
Error :inlEquality_alt,
sqequalBase,
equalitySymmetry,
Error :inhabitedIsType,
unionElimination,
equalityTransitivity,
independent_functionElimination,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
instantiate,
cumulativity,
Error :dependent_pairEquality_alt,
axiomEquality,
unionEquality,
functionEquality,
Error :functionIsTypeImplies,
universeEquality
Latex:
\mforall{}[F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}]
(\mdownarrow{}\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} (\mBbbN{}n?)
\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}
((\mexists{}n:\mBbbN{}. ((M n f) = (inl (F f)))) \mwedge{} (\mforall{}m:\mBbbN{}. ((\muparrow{}isl(M m f)) {}\mRightarrow{} ((M m f) = (inl (F f)))))))
Date html generated:
2019_06_20-PM-02_51_39
Last ObjectModification:
2018_11_23-PM-05_21_40
Theory : continuity
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