Nuprl Lemma : oalist_car_properties
∀a:LOSet. ∀b:AbDMon. ∀ws:|oal(a;b)|. ((↑sd_ordered(map(λx.(fst(x));ws))) ∧ (¬↑(e ∈b map(λx.(snd(x));ws))))
Proof
Definitions occuring in Statement :
oalist: oal(a;b),
sd_ordered: sd_ordered(as),
mem: a ∈b as,
map: map(f;as),
assert: ↑b,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
lambda: λx.A[x],
dset_of_mon: g↓set,
abdmonoid: AbDMon,
grp_id: e,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
pi2: snd(t),
mon: Mon,
dmon: DMon,
prop: ℙ,
false: False,
not: ¬A,
squash: ↓T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
dset_of_mon: g↓set,
dset_list: s List,
set_prod: s × t,
dset: DSet,
subtype_rel: A ⊆r B,
abdmonoid: AbDMon,
qoset: QOSet,
poset: POSet{i},
loset: LOSet,
member: t ∈ T,
uall: ∀[x:A]. B[x],
pi1: fst(t),
set_car: |p|,
mk_dset: mk_dset(T, eq),
dset_set: dset_set,
oalist: oal(a;b),
cand: A c∧ B,
and: P ∧ Q,
all: ∀x:A. B[x]
Lemmas referenced :
sq_stable_from_decidable,
assert_wf,
sd_ordered_wf,
map_wf,
set_car_wf,
set_prod_wf,
dset_of_mon_wf,
decidable__assert,
mem_wf,
grp_id_wf,
dset_of_mon_wf0,
oalist_wf,
dset_wf,
abdmonoid_wf,
loset_wf
Rules used in proof :
voidElimination,
independent_pairFormation,
imageElimination,
baseClosed,
imageMemberEquality,
introduction,
independent_functionElimination,
lambdaEquality,
applyEquality,
because_Cache,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
productElimination,
isectElimination,
lemma_by_obid,
rename,
thin,
setElimination,
sqequalRule,
sqequalHypSubstitution,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}ws:|oal(a;b)|.
((\muparrow{}sd\_ordered(map(\mlambda{}x.(fst(x));ws))) \mwedge{} (\mneg{}\muparrow{}(e \mmember{}\msubb{} map(\mlambda{}x.(snd(x));ws))))
Date html generated:
2016_05_16-AM-08_15_31
Last ObjectModification:
2016_01_16-PM-11_58_31
Theory : polynom_2
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