Nuprl Lemma : strong-continuity-implies1
∀[F:(ℕ ⟶ ℕ) ⟶ ℕ]
(↓∃M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕ?)
∀f:ℕ ⟶ ℕ. ((↓∃n:ℕ. ((M n f) = (inl (F f)) ∈ (ℕ?))) ∧ (∀n:ℕ. (M n f) = (inl (F f)) ∈ (ℕ?) supposing ↑isl(M n f))))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
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 :
cand: A c∧ B,
squash: ↓T,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
uimplies: b supposing a,
so_apply: x[s],
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
guard: {T}
Lemmas referenced :
isl_wf,
assert_wf,
isect_wf,
subtype_rel_self,
false_wf,
int_seg_subtype_nat,
subtype_rel_dep_function,
equal_wf,
squash_wf,
all_wf,
unit_wf2,
int_seg_wf,
nat_wf,
exists_wf,
squash-from-quotient,
strong-continuity2-no-inner-squash,
implies-quotient-true
Rules used in proof :
promote_hyp,
dependent_pairFormation,
productElimination,
baseClosed,
imageMemberEquality,
imageElimination,
independent_functionElimination,
inlEquality,
lambdaFormation,
independent_pairFormation,
independent_isectElimination,
functionExtensionality,
applyEquality,
productEquality,
lambdaEquality,
sqequalRule,
unionEquality,
because_Cache,
rename,
setElimination,
natural_numberEquality,
hypothesis,
functionEquality,
isectElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
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)))) \mwedge{} (\mforall{}n:\mBbbN{}. (M n f) = (inl (F f)) supposing \muparrow{}isl(M n f))))
Date html generated:
2018_05_21-PM-01_17_47
Last ObjectModification:
2018_05_18-PM-04_03_51
Theory : continuity
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