Nuprl Lemma : classical-markov
∀P:ℕ ⟶ ℙ. ((∀n:ℕ. Dec(P[n])) ⇒ (¬(∀n:ℕ. (¬P[n]))) ⇒ (∃n:ℕ. P[n]))
Proof
Definitions occuring in Statement :
nat: ℕ,
decidable: Dec(P),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x]
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],
so_apply: x[s],
classical: {P},
not: ¬A,
false: False,
exists: ∃x:A. B[x],
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
uimplies: b supposing a,
and: P ∧ Q
Lemmas referenced :
mu-dec_wf,
it_wf,
unit_wf2,
mu-dec-property,
exists_wf,
decidable_wf,
nat_wf,
all_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
functionEquality,
cumulativity,
universeEquality,
introduction,
independent_functionElimination,
dependent_pairFormation,
voidElimination,
classicalIntroduction,
rename,
setElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination
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]))
Date html generated:
2016_05_14-AM-07_30_25
Last ObjectModification:
2016_01_14-PM-09_58_04
Theory : int_2
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