### 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

Home Index