Nuprl Lemma : decidable__tree-big
∀[T:Type]. ∀[A:(T List) ⟶ ℙ]. ((∃k:ℕ. T ~ ℕk) ⇒ Decidable(A) ⇒ (∀n:ℕ. Dec(tree-big(T;A;n))))
Proof
Definitions occuring in Statement :
tree-big: tree-big(T;A;n),
dec-predicate: Decidable(X),
equipollent: A ~ B,
list: T List,
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
tree-big: tree-big(T;A;n),
dec-predicate: Decidable(X),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
nat: ℕ
Lemmas referenced :
decidable__all_length,
list_wf,
nat_wf,
all_wf,
decidable_wf,
exists_wf,
equipollent_wf,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
natural_numberEquality,
setElimination,
rename,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[A:(T List) {}\mrightarrow{} \mBbbP{}]. ((\mexists{}k:\mBbbN{}. T \msim{} \mBbbN{}k) {}\mRightarrow{} Decidable(A) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. Dec(tree-big(T;A;n))))
Date html generated:
2016_05_14-PM-04_10_08
Last ObjectModification:
2015_12_26-PM-07_54_23
Theory : fan-theorem
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