### Nuprl Lemma : nat-star-retract-property

`∀s:ℕ ⟶ ℕ. (∃n:ℕ. 0 < s n `⇐⇒` ∃n:ℕ. 0 < nat-star-retract(s) n)`

Proof

Definitions occuring in Statement :  nat-star-retract: `nat-star-retract(s)` nat: `ℕ` less_than: `a < b` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` exists: `∃x:A. B[x]` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` rev_implies: `P `` Q` nat-star: `ℕ*` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` decidable: `Dec(P)` or: `P ∨ Q` le: `A ≤ B` less_than': `less_than'(a;b)` nat: `ℕ` ge: `i ≥ j ` nat-star-retract: `nat-star-retract(s)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` less_than: `a < b` squash: `↓T`
Lemmas referenced :  exists_wf nat_wf less_than_wf nat-star-retract_wf nat-star_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 int_seg_wf decidable__equal_int subtract_wf int_seg_subtype false_wf decidable__le intformnot_wf itermSubtract_wf intformeq_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_formula_prop_eq_lemma le_wf all_wf int_seg_subtype_nat decidable__lt lelt_wf set_wf primrec-wf2 nat_properties itermAdd_wf int_term_value_add_lemma decidable__exists_int_seg bl-exists_wf upto_wf l_member_wf lt_int_wf bool_wf eqtt_to_assert assert-bl-exists l_exists_functionality assert_wf iff_weakening_uiff subtype_rel_set assert_of_lt_int eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot l_exists_wf l_exists_iff bnot_wf not_wf bool_cases iff_transitivity assert_of_bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation independent_pairFormation sqequalHypSubstitution productElimination thin cut introduction extract_by_obid isectElimination hypothesis sqequalRule lambdaEquality natural_numberEquality applyEquality functionExtensionality hypothesisEquality because_Cache setElimination rename functionEquality independent_isectElimination dependent_pairFormation int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality computeAll unionElimination addLevel equalityTransitivity equalitySymmetry applyLambdaEquality levelHypothesis hypothesis_subsumption dependent_set_memberEquality addEquality independent_functionElimination instantiate setEquality equalityElimination promote_hyp cumulativity imageElimination impliesFunctionality

Latex:
\mforall{}s:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  (\mexists{}n:\mBbbN{}.  0  <  s  n  \mLeftarrow{}{}\mRightarrow{}  \mexists{}n:\mBbbN{}.  0  <  nat-star-retract(s)  n)

Date html generated: 2017_04_17-AM-09_55_20
Last ObjectModification: 2017_02_27-PM-05_49_45

Theory : continuity

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