Nuprl Lemma : general-uniform-continuity-from-fan-ext
∀[B:ℕ ⟶ Type]
⇃(∀i:ℕ. ∀K:B[i] ⟶ ℕ. (∃Bnd:ℕ [(∀t:B[i]. ((K t) ≤ Bnd))]))
⇒ (∀[T:Type]
∀F:(i:ℕ ⟶ B[i]) ⟶ T
(⇃(∃M:n:ℕ ⟶ (i:ℕn ⟶ B[i]) ⟶ (T?) [(∀f:i:ℕ ⟶ B[i]
((∃n:ℕ. ((M n f) = (inl (F f)) ∈ (T?)))
∧ (∀n:ℕ. (M n f) = (inl (F f)) ∈ (T?) supposing ↑isl(M n f))))])
⇒ ⇃(∃n:ℕ. ∀f,g:i:ℕ ⟶ B[i]. ((f = g ∈ (i:ℕn ⟶ B[i])) ⇒ ((F f) = (F g) ∈ T)))))
supposing ∀i:ℕ. B[i]
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
isl: isl(x),
btrue: tt,
it: ⋅,
bfalse: ff,
subtract: n - m,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
bool: 𝔹,
unit: Unit,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
general-uniform-continuity-from-fan,
implies-quotient-true2,
trivial-quotient-true,
simple_more_general_fan_theorem-ext,
decidable__assert,
implies-quotient-true,
so_lambda: so_lambda4,
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
general-uniform-continuity-from-fan,
subtype_base_sq,
bool_wf,
bool_subtype_base,
unit_wf2,
lifting-strict-decide,
strict4-decide,
implies-quotient-true2,
trivial-quotient-true,
simple_more_general_fan_theorem-ext,
decidable__assert,
implies-quotient-true
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
cumulativity,
independent_isectElimination,
inlEquality_alt,
closedConclusion,
axiomEquality,
natural_numberEquality,
universeIsType,
dependent_functionElimination,
independent_functionElimination,
baseClosed,
Error :memTop
Latex:
\mforall{}[B:\mBbbN{} {}\mrightarrow{} Type]
\00D9(\mforall{}i:\mBbbN{}. \mforall{}K:B[i] {}\mrightarrow{} \mBbbN{}. (\mexists{}Bnd:\mBbbN{} [(\mforall{}t:B[i]. ((K t) \mleq{} Bnd))]))
{}\mRightarrow{} (\mforall{}[T:Type]
\mforall{}F:(i:\mBbbN{} {}\mrightarrow{} B[i]) {}\mrightarrow{} T
(\00D9(\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (i:\mBbbN{}n {}\mrightarrow{} B[i]) {}\mrightarrow{} (T?) [(\mforall{}f:i:\mBbbN{} {}\mrightarrow{} B[i]
((\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))))])
{}\mRightarrow{} \00D9(\mexists{}n:\mBbbN{}. \mforall{}f,g:i:\mBbbN{} {}\mrightarrow{} B[i]. ((f = g) {}\mRightarrow{} ((F f) = (F g))))))
supposing \mforall{}i:\mBbbN{}. B[i]
Date html generated:
2020_05_19-PM-10_04_52
Last ObjectModification:
2019_12_31-PM-00_57_08
Theory : continuity
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