Nuprl Lemma : not-l_all-dec
∀[T:Type]. ∀L:T List. ∀P:T ⟶ ℙ. ((∀x:T. Dec(P[x])) ⇒ (¬(∀x∈L.P[x]) ⇐⇒ (∃x∈L. ¬P[x])))
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
decidable: Dec(P),
or: P ∨ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
not_wf,
l_all_wf,
l_member_wf,
l_exists_wf,
all_wf,
decidable_wf,
list_wf,
not-l_exists,
l_all_iff,
decidable__l_exists,
decidable__not,
l_exists_iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
setEquality,
hypothesis,
independent_functionElimination,
voidElimination,
functionEquality,
cumulativity,
universeEquality,
because_Cache,
dependent_functionElimination,
productElimination,
unionElimination
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}P:T {}\mrightarrow{} \mBbbP{}. ((\mforall{}x:T. Dec(P[x])) {}\mRightarrow{} (\mneg{}(\mforall{}x\mmember{}L.P[x]) \mLeftarrow{}{}\mRightarrow{} (\mexists{}x\mmember{}L. \mneg{}P[x])))
Date html generated:
2016_05_14-AM-07_48_07
Last ObjectModification:
2015_12_26-PM-02_55_35
Theory : list_1
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