### Nuprl Lemma : not-ni-eventually-equal-inf

`∀[x:ℕ∞]. (¬ni-eventually-equal(x;0∞) `⇐⇒` x = ∞ ∈ ℕ∞)`

Proof

Definitions occuring in Statement :  ni-eventually-equal: `ni-eventually-equal(f;g)` nat-inf-infinity: `∞` nat2inf: `n∞` nat-inf: `ℕ∞` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` not: `¬A` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` prop: `ℙ` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` rev_implies: `P `` Q` nat-inf: `ℕ∞` so_lambda: `λ2x.t[x]` ge: `i ≥ j ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` so_apply: `x[s]` nat-inf-infinity: `∞` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` ni-eventually-equal: `ni-eventually-equal(f;g)` subtype_rel: `A ⊆r B` nat2inf: `n∞` int_upper: `{i...}` subtract: `n - m`
Lemmas referenced :  not_wf ni-eventually-equal_wf nat2inf_wf false_wf le_wf equal-wf-T-base nat-inf_wf nat_wf all_wf assert_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_wf bool_wf eqtt_to_assert btrue_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot int_upper_wf int_upper_subtype_nat lt_int_wf assert_of_lt_int int_upper_properties intformless_wf int_formula_prop_less_lemma less_than_wf bfalse_wf ge_wf add-zero subtract_wf itermSubtract_wf int_term_value_subtract_lemma add-associates subtract-add-cancel assert_elim minus-one-mul add-commutes add-mul-special zero-mul zero-add not_assert_elim int_subtype_base btrue_neq_bfalse equal-wf-base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation lambdaFormation hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality dependent_set_memberEquality natural_numberEquality sqequalRule independent_functionElimination voidElimination because_Cache baseClosed productElimination independent_pairEquality lambdaEquality dependent_functionElimination axiomEquality setElimination rename functionExtensionality functionEquality applyEquality addEquality unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidEquality computeAll equalityElimination equalityTransitivity equalitySymmetry promote_hyp instantiate cumulativity intWeakElimination hyp_replacement applyLambdaEquality

Latex:
\mforall{}[x:\mBbbN{}\minfty{}].  (\mneg{}ni-eventually-equal(x;0\minfty{})  \mLeftarrow{}{}\mRightarrow{}  x  =  \minfty{})

Date html generated: 2017_10_01-AM-08_30_20
Last ObjectModification: 2017_07_26-PM-04_24_31

Theory : basic

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