Nuprl Lemma : strong-continuity-implies1-sp
∀[F:(ℕ ⟶ ℕ) ⟶ ℕ]. (↓∃M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕ?). ∀f:ℕ ⟶ ℕ. (↓∃n:ℕ. ((M n f) = (inl (F f)) ∈ (ℕ?))))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
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,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q
Lemmas referenced :
subtype_rel_self,
false_wf,
int_seg_subtype_nat,
int_seg_wf,
subtype_rel_dep_function,
unit_wf2,
equal_wf,
exists_wf,
squash_wf,
all_wf,
strong-continuity-implies1,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalHypSubstitution,
imageElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
functionEquality,
lemma_by_obid,
isectElimination,
productElimination,
dependent_pairFormation,
lambdaFormation,
dependent_functionElimination,
because_Cache,
lambdaEquality,
unionEquality,
applyEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
inlEquality
Latex:
\mforall{}[F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}]. (\mdownarrow{}\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} (\mBbbN{}?). \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. (\mdownarrow{}\mexists{}n:\mBbbN{}. ((M n f) = (inl (F f)))))
Date html generated:
2016_05_14-PM-09_41_51
Last ObjectModification:
2016_01_15-PM-10_55_49
Theory : continuity
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