Nuprl Lemma : not-not-l_all-xmiddle
∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ ℙ]. (¬¬(∀x∈L.P[x] ∨ (¬P[x])))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
not: ¬A,
or: P ∨ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
uimplies: b supposing a,
sq_stable: SqStable(P),
lelt: i ≤ j < k,
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
l_all: (∀x∈L.P[x]),
or: P ∨ Q
Lemmas referenced :
not-not-all-int_seg-xmiddle,
length_wf,
select_wf,
sq_stable__le,
select_member,
l_member_wf,
int_seg_wf,
l_all_wf,
not_wf,
istype-void,
list_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
thin,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
natural_numberEquality,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
applyEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
because_Cache,
independent_isectElimination,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
universeIsType,
voidElimination,
functionIsType,
unionEquality,
setIsType,
functionIsTypeImplies,
inhabitedIsType,
universeEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]. (\mneg{}\mneg{}(\mforall{}x\mmember{}L.P[x] \mvee{} (\mneg{}P[x])))
Date html generated:
2020_05_19-PM-09_37_13
Last ObjectModification:
2019_10_31-PM-10_12_26
Theory : list_0
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