Nuprl Lemma : strong-continuity-rel

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

Proof

Definitions occuring in Statement :  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]` 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` implies: `P `` Q` 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` so_apply: `x[s]` exists: `∃x:A. B[x]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` int_seg: `{i..j-}` lelt: `i ≤ j < k` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` guard: `{T}` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` cand: `A 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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