Nuprl Lemma : compact-nat-inf
∀p:ℕ∞ ⟶ 𝔹. ((∃x:ℕ∞. p x = ff) ∨ (∀x:ℕ∞. p x = tt))
Proof
Definitions occuring in Statement :
nat-inf: ℕ∞,
bfalse: ff,
btrue: tt,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
or: P ∨ Q,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
assert: ↑b,
ifthenelse: if b then t else f fi ,
true: True,
rev_implies: P ⇐ Q
Lemmas referenced :
nat-inf_wf,
ni-selector_wf,
bool_wf,
equal-wf-T-base,
all_wf,
equal_wf,
btrue_neq_bfalse,
ni-selector-property,
exists_wf,
not_wf,
iff_imp_equal_bool,
false_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
functionExtensionality,
hypothesisEquality,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
unionElimination,
equalityElimination,
inlFormation,
dependent_pairFormation,
baseClosed,
sqequalRule,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
functionEquality,
voidElimination,
addLevel,
impliesFunctionality,
productElimination,
inrFormation,
because_Cache,
independent_isectElimination,
independent_pairFormation,
natural_numberEquality
Latex:
\mforall{}p:\mBbbN{}\minfty{} {}\mrightarrow{} \mBbbB{}. ((\mexists{}x:\mBbbN{}\minfty{}. p x = ff) \mvee{} (\mforall{}x:\mBbbN{}\minfty{}. p x = tt))
Date html generated:
2017_10_01-AM-08_29_30
Last ObjectModification:
2017_07_26-PM-04_24_02
Theory : basic
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