### Nuprl Lemma : uniform-continuity-from-fan-ext

`∀[T:Type]`
`  ∀F:(ℕ ⟶ 𝔹) ⟶ T`
`    (⇃(∃M:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ (T?) [(∀f:ℕ ⟶ 𝔹`
`                                       ((∃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:ℕ ⟶ 𝔹.  ((f = g ∈ (ℕn ⟶ 𝔹)) `` ((F f) = (F g) ∈ T))))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isl: `isl(x)` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` 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 ` ucont: `ucont(F;M)` uniform-continuity-from-fan implies-quotient-true2 trivial-quotient-true fan_theorem-ext decidable__assert implies-quotient-true uall: `∀[x:A]. B[x]` so_lambda: `so_lambda(x,y,z,w.t[x; y; z; w])` so_apply: `x[s1;s2;s3;s4]` so_lambda: `λ2x y.t[x; y]` top: `Top` so_apply: `x[s1;s2]` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  uniform-continuity-from-fan lifting-strict-callbyvalue istype-void strict4-spread lifting-strict-decide strict4-decide implies-quotient-true2 trivial-quotient-true 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 baseClosed Error :isect_memberEquality_alt,  voidElimination independent_isectElimination

Latex:
\mforall{}[T:Type]
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  T
(\00D9(\mexists{}M:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  (T?)  [(\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}
((\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:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.    ((f  =  g)  {}\mRightarrow{}  ((F  f)  =  (F  g)))))

Date html generated: 2019_06_20-PM-02_52_36
Last ObjectModification: 2019_03_12-PM-06_00_00

Theory : continuity

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