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

`∀[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 :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` nat: `ℕ` so_lambda: `λ2x.t[x]` prop: `ℙ` and: `P ∧ Q` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` isl: `isl(x)` so_apply: `x[s]` sq_exists: `∃x:A [B[x]]` so_apply: `x[s1;s2]` squash: `↓T` guard: `{T}` so_lambda: `λ2x y.t[x; y]` true: `True` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` cand: `A c∧ B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` sq_stable: `SqStable(P)` sq_type: `SQType(T)` outl: `outl(x)`
Lemmas referenced :  implies-quotient-true2 sq_exists_wf nat_wf int_seg_wf bool_wf unit_wf2 equal_wf assert_wf istype-nat trivial-quotient-true fan_theorem-ext btrue_wf bfalse_wf istype-assert quotient_wf true_wf equiv_rel_true istype-universe assert_functionality_wrt_uiff isl_wf istype-void squash_wf subtype_rel_function int_seg_subtype_nat istype-false subtype_rel_self decidable__assert int_seg_properties nat_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf intformle_wf istype-int int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_le_lemma int_formula_prop_wf subtract_wf decidable__le itermSubtract_wf int_term_value_subtract_lemma istype-le int_seg_subtype not-le-2 condition-implies-le minus-one-mul add-commutes minus-one-mul-top minus-add minus-minus add-associates add-swap zero-add sq_stable__le less-iff-le add_functionality_wrt_le le-add-cancel iff_weakening_equal outl_wf subtype_base_sq bool_subtype_base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  Error :lambdaFormation_alt,  cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination functionEquality hypothesis natural_numberEquality setElimination rename because_Cache unionEquality hypothesisEquality sqequalRule Error :lambdaEquality_alt,  productEquality applyEquality Error :inlEquality_alt,  Error :universeIsType,  isectEquality Error :inhabitedIsType,  unionElimination Error :equalityIstype,  equalityTransitivity equalitySymmetry dependent_functionElimination independent_functionElimination Error :functionIsType,  Error :unionIsType,  independent_isectElimination imageElimination imageMemberEquality baseClosed Error :setIsType,  Error :productIsType,  Error :isectIsType,  instantiate universeEquality productElimination Error :dependent_pairFormation_alt,  voidEquality independent_pairFormation approximateComputation int_eqEquality Error :isect_memberEquality_alt,  voidElimination Error :dependent_set_memberEquality_alt,  addEquality minusEquality applyLambdaEquality cumulativity

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_23
Last ObjectModification: 2019_01_27-PM-01_57_32

Theory : continuity

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