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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