Nuprl Lemma : weak-Markov-principle2

`∀a:ℕ*. ((∀c:ℕ*. ((¬¬(∃n:ℕ. (¬((a n) = (c n) ∈ ℤ)))) ∨ (¬¬(∃n:ℕ. (¬(0 = (c n) ∈ ℤ)))))) `` (∃n:ℕ. 0 < a n))`

Proof

Definitions occuring in Statement :  nat-star: `ℕ*` nat: `ℕ` less_than: `a < b` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` or: `P ∨ Q` apply: `f a` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` nat-star: `ℕ*` subtype_rel: `A ⊆r B` nat: `ℕ` so_apply: `x[s]` or: `P ∨ Q` cand: `A c∧ B` not: `¬A` false: `False` less_than': `less_than'(a;b)` and: `P ∧ Q` le: `A ≤ B` exists: `∃x:A. B[x]` true: `True` guard: `{T}` sq_type: `SQType(T)` uimplies: `b supposing a` pi1: `fst(t)` decidable: `Dec(P)` squash: `↓T` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` nat-star-0: `0` top: `Top` satisfiable_int_formula: `satisfiable_int_formula(fmla)` ge: `i ≥ j ` lelt: `i ≤ j < k` int_seg: `{i..j-}` so_apply: `x[s1;s2]` so_lambda: `λ2x y.t[x; y]`
Lemmas referenced :  all_wf nat-star_wf or_wf not_wf exists_wf nat_wf equal_wf equal-wf-base-T nat-star-retract_wf equal-wf-T-base equal-wf-base le_wf false_wf int_subtype_base subtype_base_sq strong-continuity2-implies-weak decidable__equal_int nat-star-0_wf squash_wf true_wf nat-star-retract-id subtype_rel_self iff_weakening_equal int_seg_subtype_nat int_seg_wf subtype_rel_dep_function quotient-implies-squash int_formula_prop_wf int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermConstant_wf itermVar_wf intformeq_wf intformnot_wf intformand_wf satisfiable-full-omega-tt nat_properties decidable__not decidable__exists_int_seg decidable__equal_nat int_formula_prop_le_lemma intformle_wf decidable__le int_seg_properties unit_wf2 mu-dec-property it_wf less_than_wf int_formula_prop_less_lemma intformless_wf decidable__lt mu-dec_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis sqequalRule lambdaEquality intEquality applyEquality setElimination rename hypothesisEquality because_Cache baseClosed functionEquality unionElimination functionExtensionality dependent_functionElimination productEquality voidElimination independent_functionElimination independent_pairFormation natural_numberEquality dependent_set_memberEquality dependent_pairFormation promote_hyp levelHypothesis equalitySymmetry equalityTransitivity independent_isectElimination cumulativity instantiate addLevel productElimination imageElimination universeEquality imageMemberEquality computeAll voidEquality isect_memberEquality int_eqEquality applyLambdaEquality

Latex:
\mforall{}a:\mBbbN{}*.  ((\mforall{}c:\mBbbN{}*.  ((\mneg{}\mneg{}(\mexists{}n:\mBbbN{}.  (\mneg{}((a  n)  =  (c  n)))))  \mvee{}  (\mneg{}\mneg{}(\mexists{}n:\mBbbN{}.  (\mneg{}(0  =  (c  n)))))))  {}\mRightarrow{}  (\mexists{}n:\mBbbN{}.  0  <  a  n))

Date html generated: 2018_05_21-PM-01_19_02
Last ObjectModification: 2018_05_15-PM-04_32_46

Theory : continuity

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