Nuprl Lemma : decidable__exists_length
∀[T:Type]. ∀[P:(T List) ⟶ ℙ].
((∀L:T List. Dec(P[L])) ⇒ (∃k:ℕ. T ~ ℕk) ⇒ (∀n:ℕ. Dec(∃L:T List. ((||L|| = n ∈ ℤ) ∧ P[L]))))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
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],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
equipollent: A ~ B,
biject: Bij(A;B;f),
and: P ∧ Q,
surject: Surj(A;B;f),
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
pi1: fst(t),
cand: A c∧ B,
not: ¬A,
false: False,
inject: Inj(A;B;f)
Lemmas referenced :
nat_wf,
exists_wf,
equipollent_wf,
int_seg_wf,
all_wf,
list_wf,
decidable_wf,
equipollent-list,
decidable__exists_int_seg,
exp_wf2,
equal_wf,
length_wf,
not_wf,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
promote_hyp,
instantiate,
because_Cache,
setEquality,
intEquality,
unionElimination,
inlFormation,
productEquality,
inrFormation,
dependent_set_memberEquality,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
voidElimination,
addLevel,
hyp_replacement,
levelHypothesis
Latex:
\mforall{}[T:Type]. \mforall{}[P:(T List) {}\mrightarrow{} \mBbbP{}].
((\mforall{}L:T List. Dec(P[L])) {}\mRightarrow{} (\mexists{}k:\mBbbN{}. T \msim{} \mBbbN{}k) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. Dec(\mexists{}L:T List. ((||L|| = n) \mwedge{} P[L]))))
Date html generated:
2017_04_17-AM-09_36_04
Last ObjectModification:
2017_02_27-PM-05_34_11
Theory : equipollence!!cardinality!
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