### Nuprl Lemma : strong-continuity-rel-unique-pair

`∀P:(ℕ ⟶ ℕ) ⟶ ℕ ⟶ ℙ. ∀F:∀f:ℕ ⟶ ℕ. ⇃(∃n:ℕ. (P f n)).`
`  ⇃(∃M:n:ℕ ⟶ s:(ℕn ⟶ ℕ) ⟶ (k:ℕn × (P ext2Baire(n;s;0) k)?)`
`     ∀f:ℕ ⟶ ℕ`
`       ∃n:ℕ`
`        ∃k:ℕn`
`         ∃p:P ext2Baire(n;f;0) k`
`          (((M n f) = (inl <k, p>) ∈ (k:ℕn × (P ext2Baire(n;f;0) k)?)) ∧ (∀m:ℕ. ((↑isl(M m f)) `` (m = n ∈ ℕ)))))`

Proof

Definitions occuring in Statement :  ext2Baire: `ext2Baire(n;f;d)` quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isl: `isl(x)` prop: `ℙ` all: `∀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]` pair: `<a, b>` product: `x:A × B[x]` inl: `inl x` union: `left + right` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` so_lambda: `λ2x.t[x]` prop: `ℙ` and: `P ∧ Q` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` so_apply: `x[s]` exists: `∃x:A. B[x]` guard: `{T}` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` spreadn: spread4 exposed-it: `exposed-it` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` int_seg: `{i..j-}` lelt: `i ≤ j < k` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` cand: `A c∧ B` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` isl: `isl(x)` ext2Baire: `ext2Baire(n;f;d)` true: `True` nequal: `a ≠ b ∈ T `
Lemmas referenced :  strong-continuity-rel-unique implies-quotient-true exists_wf int_seg_wf unit_wf2 all_wf int_seg_subtype_nat false_wf equal_wf subtype_rel_dep_function nat_wf subtype_rel_self subtype_rel_union assert_wf isl_wf ext2Baire_wf le_wf quotient_wf true_wf equiv_rel_true eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int nat_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_wf lelt_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int int_seg_properties btrue_wf bfalse_wf lt_int_wf assert_of_lt_int less_than_wf decidable__equal_int and_wf set_subtype_base int_subtype_base union_subtype_base unit_subtype_base decidable__le intformle_wf itermConstant_wf int_formula_prop_le_lemma int_term_value_constant_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality isectElimination functionEquality because_Cache natural_numberEquality setElimination rename hypothesis unionEquality sqequalRule lambdaEquality productEquality applyEquality functionExtensionality independent_isectElimination independent_pairFormation inlEquality dependent_set_memberEquality dependent_pairEquality independent_functionElimination productElimination cumulativity universeEquality dependent_pairFormation unionElimination equalityElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll equalityTransitivity equalitySymmetry promote_hyp instantiate inrEquality axiomEquality applyLambdaEquality

Latex:
\mforall{}P:(\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbN{}  {}\mrightarrow{}  \mBbbP{}.  \mforall{}F:\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mexists{}n:\mBbbN{}.  (P  f  n)).
\00D9(\mexists{}M:n:\mBbbN{}  {}\mrightarrow{}  s:(\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  (k:\mBbbN{}n  \mtimes{}  (P  ext2Baire(n;s;0)  k)?)
\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}
\mexists{}n:\mBbbN{}
\mexists{}k:\mBbbN{}n.  \mexists{}p:P  ext2Baire(n;f;0)  k.  (((M  n  f)  =  (inl  <k,  p>))  \mwedge{}  (\mforall{}m:\mBbbN{}.  ((\muparrow{}isl(M  m  f))  {}\mRightarrow{}  (m  =  n)\000C))))

Date html generated: 2017_04_17-AM-10_02_44
Last ObjectModification: 2017_02_27-PM-05_54_46

Theory : continuity

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