Nuprl Lemma : weak-Markov-principle
∀a,b:ℕ ⟶ ℕ.
((∀c:ℕ ⟶ ℕ. ((¬¬(∃n:ℕ. (¬((a n) = (c n) ∈ ℤ)))) ∨ (¬¬(∃n:ℕ. (¬((b n) = (c n) ∈ ℤ))))))
⇒ (∃n:ℕ. (¬((a n) = (b n) ∈ ℤ))))
Proof
Definitions occuring in Statement :
nat: ℕ,
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],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
squash: ↓T,
exists: ∃x:A. B[x],
true: True,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
guard: {T},
uall: ∀[x:A]. B[x],
prop: ℙ,
false: False,
not: ¬A,
or: P ∨ Q,
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
iff_weakening_equal,
true_wf,
squash_wf,
exists_wf,
or_wf,
all_wf,
not_wf,
nat_wf,
equal_wf,
weak-Markov-principle-alt
Rules used in proof :
independent_isectElimination,
baseClosed,
imageMemberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
productElimination,
natural_numberEquality,
intEquality,
lambdaEquality,
because_Cache,
inrFormation,
sqequalRule,
applyEquality,
functionExtensionality,
functionEquality,
isectElimination,
voidElimination,
inlFormation,
unionElimination,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}a,b:\mBbbN{} {}\mrightarrow{} \mBbbN{}.
((\mforall{}c:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((\mneg{}\mneg{}(\mexists{}n:\mBbbN{}. (\mneg{}((a n) = (c n))))) \mvee{} (\mneg{}\mneg{}(\mexists{}n:\mBbbN{}. (\mneg{}((b n) = (c n)))))))
{}\mRightarrow{} (\mexists{}n:\mBbbN{}. (\mneg{}((a n) = (b n)))))
Date html generated:
2017_09_29-PM-06_06_46
Last ObjectModification:
2017_08_30-PM-00_28_08
Theory : continuity
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