### Nuprl Lemma : markov-streamless-iff-not-not-enum

`(∀P:ℕ ⟶ ℙ. ((∀m:ℕ. ((P m) ∨ (¬(P m)))) `` (¬(∀m:ℕ. (¬(P m)))) `` (∃m:ℕ. (P m))))`
` (∀T:Type. (streamless(T) `⇐⇒` (∀x,y:T.  Dec(x = y ∈ T)) ∧ (¬¬(∃L:T List. ∀x:T. (x ∈ L)))))`

Proof

Definitions occuring in Statement :  streamless: `streamless(T)` l_member: `(x ∈ l)` list: `T List` nat: `ℕ` decidable: `Dec(P)` prop: `ℙ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` not: `¬A` implies: `P `` Q` or: `P ∨ Q` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  implies: `P `` Q` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` uall: `∀[x:A]. B[x]` member: `t ∈ T` not: `¬A` false: `False` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` streamless: `streamless(T)` nat: `ℕ` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` cand: `A c∧ B` decidable: `Dec(P)` or: `P ∨ Q` guard: `{T}` l_member: `(x ∈ l)` int_seg: `{i..j-}` lelt: `i ≤ j < k` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` less_than: `a < b` squash: `↓T` pi1: `fst(t)` inject: `Inj(A;B;f)` true: `True` uiff: `uiff(P;Q)`
Lemmas referenced :  streamless-dec-equal streamless-implies-not-not-enum not_wf exists_wf list_wf all_wf l_member_wf streamless_wf decidable_wf equal_wf nat_wf or_wf int_seg_wf int_seg_subtype_nat false_wf decidable__exists_int_seg decidable__cand decidable__not decidable__equal_nat lelt_wf length_wf select_wf int_seg_properties nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_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_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma non_neg_length le_wf length_wf_nat squash_wf true_wf iff_weakening_equal less_than_wf intformeq_wf int_formula_prop_eq_lemma decidable__equal_int pigeon-hole add_nat_wf add-is-int-iff itermAdd_wf int_term_value_add_lemma subtype_rel_dep_function subtype_rel_self
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation independent_pairFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache independent_functionElimination hypothesis dependent_functionElimination hypothesisEquality voidElimination cumulativity sqequalRule lambdaEquality productElimination productEquality universeEquality instantiate functionEquality applyEquality functionExtensionality natural_numberEquality setElimination rename independent_isectElimination isect_memberEquality unionElimination inlFormation inrFormation dependent_pairFormation dependent_set_memberEquality equalityTransitivity equalitySymmetry int_eqEquality intEquality voidEquality computeAll imageElimination promote_hyp applyLambdaEquality imageMemberEquality baseClosed addEquality pointwiseFunctionality baseApply closedConclusion

Latex:
(\mforall{}P:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}.  ((\mforall{}m:\mBbbN{}.  ((P  m)  \mvee{}  (\mneg{}(P  m))))  {}\mRightarrow{}  (\mneg{}(\mforall{}m:\mBbbN{}.  (\mneg{}(P  m))))  {}\mRightarrow{}  (\mexists{}m:\mBbbN{}.  (P  m))))
{}\mRightarrow{}  (\mforall{}T:Type.  (streamless(T)  \mLeftarrow{}{}\mRightarrow{}  (\mforall{}x,y:T.    Dec(x  =  y))  \mwedge{}  (\mneg{}\mneg{}(\mexists{}L:T  List.  \mforall{}x:T.  (x  \mmember{}  L)))))

Date html generated: 2018_05_21-PM-09_03_01
Last ObjectModification: 2017_07_26-PM-06_25_50

Theory : general

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