Nuprl Lemma : nat-strong-overt-implies-Markov
sOvert(ℕ) 
⇒ (∀g:ℕ ⟶ 𝔹. ((¬(∀n:ℕ. g n = ff)) 
⇒ (∃n:ℕ. g n = tt)))
Proof
Definitions occuring in Statement : 
strong-overt: sOvert(X)
, 
nat: ℕ
, 
bfalse: ff
, 
btrue: tt
, 
bool: 𝔹
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
strong-overt: sOvert(X)
, 
member: t ∈ T
, 
exists: ∃x:A. B[x]
, 
in-open: x ∈ A
, 
Open: Open(X)
, 
prop: ℙ
, 
uall: ∀[x:A]. B[x]
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
top: Top
, 
subtype_rel: A ⊆r B
, 
pi1: fst(t)
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
uimplies: b supposing a
, 
bfalse: ff
, 
or: P ∨ Q
, 
sq_type: SQType(T)
, 
guard: {T}
, 
bnot: ¬bb
, 
ifthenelse: if b then t else f fi 
, 
assert: ↑b
, 
false: False
, 
not: ¬A
, 
rev_implies: P 
⇐ Q
, 
true: True
Lemmas referenced : 
unit_wf2, 
not_wf, 
all_wf, 
nat_wf, 
equal-wf-T-base, 
bool_wf, 
strong-overt_wf, 
ifthenelse_wf, 
pi1_wf_top, 
Sierpinski_wf, 
Sierpinski-top_wf, 
subtype-Sierpinski, 
Sierpinski-bottom_wf, 
it_wf, 
Sierpinski-unequal, 
eqtt_to_assert, 
btrue_wf, 
eqff_to_assert, 
equal_wf, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_subtype_base, 
assert-bnot, 
exists_wf, 
btrue_neq_bfalse, 
not-Sierpinski-bottom, 
rev_implies_wf, 
bfalse_wf, 
iff_imp_equal_bool, 
assert_wf, 
true_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
cut, 
introduction, 
extract_by_obid, 
hypothesis, 
productElimination, 
sqequalRule, 
isectElimination, 
lambdaEquality, 
applyEquality, 
functionExtensionality, 
hypothesisEquality, 
baseClosed, 
functionEquality, 
independent_pairEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
productEquality, 
independent_pairFormation, 
dependent_pairFormation, 
unionElimination, 
equalityElimination, 
equalityTransitivity, 
equalitySymmetry, 
independent_isectElimination, 
promote_hyp, 
instantiate, 
cumulativity, 
independent_functionElimination, 
because_Cache, 
hyp_replacement, 
applyLambdaEquality, 
natural_numberEquality
Latex:
sOvert(\mBbbN{})  {}\mRightarrow{}  (\mforall{}g:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.  ((\mneg{}(\mforall{}n:\mBbbN{}.  g  n  =  ff))  {}\mRightarrow{}  (\mexists{}n:\mBbbN{}.  g  n  =  tt)))
Date html generated:
2019_10_31-AM-07_19_28
Last ObjectModification:
2017_07_28-AM-09_12_29
Theory : synthetic!topology
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