### Nuprl Lemma : finite-partition-property

`∀k:ℕ. ∀f:ℕ ⟶ ℕk.  (¬¬(∃i:ℕk. ∀n:ℕ. (¬¬(∃m:ℕ. (n < m ∧ ((f m) = i ∈ ℤ))))))`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` less_than: `a < b` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` false: `False` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` nat: `ℕ` so_lambda: `λ2x.t[x]` and: `P ∧ Q` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` so_apply: `x[s]` uiff: `uiff(P;Q)` uimplies: `b supposing a` exists: `∃x:A. B[x]` le: `A ≤ B` less_than': `less_than'(a;b)` guard: `{T}` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` ge: `i ≥ j ` cand: `A c∧ B` sq_type: `SQType(T)` ifthenelse: `if b then t else f fi ` btrue: `tt` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bfalse: `ff`
Lemmas referenced :  all_wf int_seg_wf not_wf nat_wf exists_wf less_than_wf equal_wf not_over_exists finite-double-negation-shift false_wf subtract_wf set_wf primrec-wf2 le_wf int_seg_properties satisfiable-full-omega-tt intformand_wf intformless_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf decidable__lt intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma lelt_wf nat_properties decidable__le ifthenelse_wf lt_int_wf assert_wf bnot_wf bool_cases subtype_base_sq bool_wf bool_subtype_base eqtt_to_assert assert_of_lt_int eqff_to_assert iff_transitivity iff_weakening_uiff assert_of_bnot decidable__equal_int intformeq_wf int_formula_prop_eq_lemma equal-wf-T-base int_subtype_base itermAdd_wf int_term_value_add_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination natural_numberEquality setElimination rename because_Cache hypothesis sqequalRule lambdaEquality productEquality intEquality applyEquality functionExtensionality hypothesisEquality addLevel impliesFunctionality productElimination independent_isectElimination independent_functionElimination voidElimination functionEquality allFunctionality levelHypothesis dependent_functionElimination equalityTransitivity equalitySymmetry instantiate dependent_pairFormation dependent_set_memberEquality independent_pairFormation int_eqEquality isect_memberEquality voidEquality computeAll unionElimination promote_hyp cumulativity baseApply closedConclusion baseClosed addEquality

Latex:
\mforall{}k:\mBbbN{}.  \mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}k.    (\mneg{}\mneg{}(\mexists{}i:\mBbbN{}k.  \mforall{}n:\mBbbN{}.  (\mneg{}\mneg{}(\mexists{}m:\mBbbN{}.  (n  <  m  \mwedge{}  ((f  m)  =  i))))))

Date html generated: 2017_10_01-AM-09_10_39
Last ObjectModification: 2017_07_26-PM-04_46_59

Theory : general

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