Nuprl Lemma : decidable-exists-iseg
∀[T:Type]. ∀[P:(T List) ⟶ ℙ]. ((∀L:T List. Dec(P[L])) ⇒ (∀L:T List. Dec(∃L':T List. (L' ≤ L ∧ P[L']))))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
list: T List,
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],
universe: Type
Definitions unfolded in proof :
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
Lemmas referenced :
decidable__exists_iseg,
list_wf,
all_wf,
decidable_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:(T List) {}\mrightarrow{} \mBbbP{}].
((\mforall{}L:T List. Dec(P[L])) {}\mRightarrow{} (\mforall{}L:T List. Dec(\mexists{}L':T List. (L' \mleq{} L \mwedge{} P[L']))))
Date html generated:
2016_05_15-PM-04_19_58
Last ObjectModification:
2015_12_27-PM-02_55_38
Theory : general
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