Nuprl Lemma : strong-continuity-rel

P:(ℕ ⟶ ℕ) ⟶ ℕ ⟶ ℙ. ∀F:∀f:ℕ ⟶ ℕ. ⇃(∃n:ℕ(P n)).
  ⇃(∃M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕn?)
     ∀f:ℕ ⟶ ℕ
       ∃n:ℕ. ∃k:ℕn. ((P k) ∧ ((M f) (inl k) ∈ (ℕ?)) ∧ (∀m:ℕ((↑isl(M f))  ((M f) (inl k) ∈ (ℕ?))))))


Proof




Definitions occuring in Statement :  quotient: x,y:A//B[x; y] int_seg: {i..j-} nat: assert: b isl: isl(x) prop: all: x:A. B[x] exists: x:A. B[x] implies:  Q and: P ∧ Q true: True unit: Unit apply: a function: x:A ⟶ B[x] inl: inl x union: left right natural_number: $n equal: t ∈ T
Definitions unfolded in proof :  all: x:A. B[x] member: t ∈ T implies:  Q uall: [x:A]. B[x] nat: so_lambda: λ2x.t[x] prop: and: P ∧ Q subtype_rel: A ⊆B uimplies: supposing a le: A ≤ B less_than': less_than'(a;b) false: False not: ¬A so_apply: x[s] exists: x:A. B[x] so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] int_seg: {i..j-} lelt: i ≤ j < k ge: i ≥  decidable: Dec(P) or: P ∨ Q guard: {T} satisfiable_int_formula: satisfiable_int_formula(fmla) top: Top cand: c∧ B true: True quotient: x,y:A//B[x; y] squash: T
Lemmas referenced :  axiom-choice-1X-quot nat_wf prop-truncation-quot exists_wf int_seg_wf unit_wf2 all_wf int_seg_subtype_nat false_wf equal_wf subtype_rel_dep_function subtype_rel_self subtype_rel_union assert_wf isl_wf quotient_wf true_wf equiv_rel_true strong-continuity2-no-inner-squash-bound less_than_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_wf lelt_wf quotient-member-eq equal-wf-base member_wf squash_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesis hypothesisEquality independent_functionElimination isectElimination functionEquality because_Cache natural_numberEquality setElimination rename unionEquality sqequalRule lambdaEquality productEquality applyEquality functionExtensionality independent_isectElimination independent_pairFormation inlEquality cumulativity universeEquality productElimination dependent_pairFormation dependent_set_memberEquality unionElimination equalityTransitivity equalitySymmetry applyLambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll promote_hyp pointwiseFunctionality pertypeElimination imageElimination imageMemberEquality baseClosed

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{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  (\mBbbN{}n?)
          \mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}
              \mexists{}n:\mBbbN{}
                \mexists{}k:\mBbbN{}n.  ((P  f  k)  \mwedge{}  ((M  n  f)  =  (inl  k))  \mwedge{}  (\mforall{}m:\mBbbN{}.  ((\muparrow{}isl(M  m  f))  {}\mRightarrow{}  ((M  m  f)  =  (inl  k))))))



Date html generated: 2017_04_17-AM-10_02_22
Last ObjectModification: 2017_02_27-PM-05_54_40

Theory : continuity


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