### Nuprl Lemma : MP+truncated-KS-imply-truncated-LEM

`(∀P:ℕ ⟶ ℙ. ((∀n:ℕ. Dec(P[n])) `` (¬(∀n:ℕ. (¬P[n]))) `` (∃n:ℕ. P[n])))`
` (∀A:ℙ. ⇃(∃a:ℕ ⟶ ℕ. (A `⇐⇒` ∃n:ℕ. ((a n) = 1 ∈ ℤ))))`
` (∀P:ℙ. ⇃(P ∨ (¬P)))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` nat: `ℕ` decidable: `Dec(P)` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` not: `¬A` implies: `P `` Q` or: `P ∨ Q` true: `True` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uiff: `uiff(P;Q)` not: `¬A` guard: `{T}` nat: `ℕ` or: `P ∨ Q` subtype_rel: `A ⊆r B` uimplies: `b supposing a` so_apply: `x[s1;s2]` so_lambda: `λ2x y.t[x; y]` and: `P ∧ Q` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` exists: `∃x:A. B[x]` so_apply: `x[s]` so_lambda: `λ2x.t[x]` uall: `∀[x:A]. B[x]` prop: `ℙ` member: `t ∈ T` all: `∀x:A. B[x]` implies: `P `` Q`
Lemmas referenced :  not_over_exists decidable__int_equal not-not-excluded-middle-quot-true implies-quotient-true2 or_wf not_wf decidable_wf equiv_rel_true true_wf equal-wf-T-base iff_wf nat_wf exists_wf quotient_wf all_wf
Rules used in proof :  natural_numberEquality promote_hyp impliesFunctionality productElimination independent_functionElimination rename setElimination dependent_functionElimination cumulativity baseClosed functionExtensionality applyEquality intEquality independent_isectElimination hypothesisEquality because_Cache hypothesis functionEquality lambdaEquality sqequalRule isectElimination sqequalHypSubstitution extract_by_obid introduction instantiate thin cut universeEquality lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
(\mforall{}P:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}.  ((\mforall{}n:\mBbbN{}.  Dec(P[n]))  {}\mRightarrow{}  (\mneg{}(\mforall{}n:\mBbbN{}.  (\mneg{}P[n])))  {}\mRightarrow{}  (\mexists{}n:\mBbbN{}.  P[n])))
{}\mRightarrow{}  (\mforall{}A:\mBbbP{}.  \00D9(\mexists{}a:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  (A  \mLeftarrow{}{}\mRightarrow{}  \mexists{}n:\mBbbN{}.  ((a  n)  =  1))))
{}\mRightarrow{}  (\mforall{}P:\mBbbP{}.  \00D9(P  \mvee{}  (\mneg{}P)))

Date html generated: 2017_04_20-AM-07_36_09
Last ObjectModification: 2017_04_11-AM-05_18_24

Theory : continuity

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