Nuprl Lemma : decidable-exists-finite
∀[T:Type]. (finite(T) ⇒ (∀[P:T ⟶ ℙ]. ((∀t:T. Dec(P[t])) ⇒ Dec(∃t:T. P[t]))))
Proof
Definitions occuring in Statement :
finite: finite(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
finite: finite(T),
exists: ∃x:A. B[x],
finite-type: finite-type(T),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
equipollent: A ~ B,
biject: Bij(A;B;f),
and: P ∧ Q
Lemmas referenced :
decidable-exists-finite-type,
finite_wf,
exists_wf,
int_seg_wf,
surject_wf,
equipollent_inversion
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
cumulativity,
universeEquality,
productElimination,
dependent_pairFormation,
functionEquality,
natural_numberEquality,
setElimination,
rename,
sqequalRule,
lambdaEquality,
because_Cache,
functionExtensionality,
applyEquality
Latex:
\mforall{}[T:Type]. (finite(T) {}\mRightarrow{} (\mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}t:T. Dec(P[t])) {}\mRightarrow{} Dec(\mexists{}t:T. P[t]))))
Date html generated:
2016_10_21-AM-11_00_13
Last ObjectModification:
2016_08_08-AM-11_27_08
Theory : equipollence!!cardinality!
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