### Nuprl Lemma : markov-streamless-function

`(∀P:ℕ ⟶ ℙ. ((∀m:ℕ. ((P m) ∨ (¬(P m)))) `` (¬(∀m:ℕ. (¬(P m)))) `` (∃m:ℕ. (P m))))`
` (∀A,B:Type.  ((∃a:ℕ. A ~ ℕa) `` streamless(B) `` streamless(A ⟶ B)))`

Proof

Definitions occuring in Statement :  streamless: `streamless(T)` equipollent: `A ~ B` int_seg: `{i..j-}` nat: `ℕ` prop: `ℙ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` or: `P ∨ Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` cand: `A c∧ B` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` nat: `ℕ` so_apply: `x[s]` subtype_rel: `A ⊆r B` exists: `∃x:A. B[x]` guard: `{T}` equipollent: `A ~ B` decidable: `Dec(P)` or: `P ∨ Q` biject: `Bij(A;B;f)` surject: `Surj(A;B;f)` squash: `↓T` true: `True` uimplies: `b supposing a` not: `¬A` false: `False`
Lemmas referenced :  markov-streamless-iff streamless_wf exists_wf nat_wf equipollent_wf int_seg_wf all_wf or_wf not_wf equipollent_inversion decidable__all_int_seg equal_wf squash_wf true_wf iff_weakening_equal and_wf exp_wf4 exp_wf2 equipollent_functionality_wrt_equipollent2 equipollent-exp equipollent_functionality_wrt_equipollent function_functionality_wrt_equipollent_right equipollent_weakening_ext-eq ext-eq_weakening function_functionality_wrt_equipollent_left
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution independent_functionElimination thin hypothesis dependent_functionElimination hypothesisEquality productElimination functionEquality cumulativity independent_pairFormation isectElimination sqequalRule lambdaEquality natural_numberEquality setElimination rename universeEquality instantiate applyEquality functionExtensionality because_Cache unionElimination inlFormation inrFormation imageElimination equalityTransitivity equalitySymmetry imageMemberEquality baseClosed independent_isectElimination dependent_set_memberEquality applyLambdaEquality voidElimination dependent_pairFormation

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{}A,B:Type.    ((\mexists{}a:\mBbbN{}.  A  \msim{}  \mBbbN{}a)  {}\mRightarrow{}  streamless(B)  {}\mRightarrow{}  streamless(A  {}\mrightarrow{}  B)))

Date html generated: 2018_05_21-PM-09_03_34
Last ObjectModification: 2017_07_26-PM-06_26_21

Theory : general

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