Nuprl Lemma : decidable__exists_length_bool
∀[P:(𝔹 List) ⟶ ℙ]. ((∀L:𝔹 List. Dec(P[L])) ⇒ (∀n:ℕ. Dec(∃L:𝔹 List. ((||L|| = n ∈ ℤ) ∧ P[L]))))
Proof
Definitions occuring in Statement :
length: ||as||,
list: T List,
nat: ℕ,
bool: 𝔹,
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,
and: P ∧ Q,
function: x:A ⟶ B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
decidable__exists_length,
bool_wf,
false_wf,
le_wf,
equipollent-two,
equipollent_wf,
int_seg_wf,
all_wf,
list_wf,
decidable_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesis,
isect_memberFormation,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
setElimination,
rename,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[P:(\mBbbB{} List) {}\mrightarrow{} \mBbbP{}]. ((\mforall{}L:\mBbbB{} List. Dec(P[L])) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. Dec(\mexists{}L:\mBbbB{} List. ((||L|| = n) \mwedge{} P[L]))))
Date html generated:
2016_05_14-PM-04_06_51
Last ObjectModification:
2015_12_26-PM-07_41_41
Theory : equipollence!!cardinality!
Home
Index