Nuprl Lemma : cons_in_oalist
∀a:LOSet. ∀b:AbDMon. ∀ws:|oal(a;b)|. ∀x:|a|. ∀y:|b|.
((↑before(x;map(λx.(fst(x));ws))) ⇒ (¬(y = e ∈ |b|)) ⇒ ([<x, y> / ws] ∈ |oal(a;b)|))
Proof
Definitions occuring in Statement :
oalist: oal(a;b),
before: before(u;ps),
map: map(f;as),
cons: [a / b],
assert: ↑b,
pi1: fst(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
lambda: λx.A[x],
pair: <a, b>,
equal: s = t ∈ T,
abdmonoid: AbDMon,
grp_id: e,
grp_car: |g|,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
subtype_rel: A ⊆r B,
dset: DSet,
set_prod: s × t,
mk_dset: mk_dset(T, eq),
set_car: |p|,
pi1: fst(t),
oalist: oal(a;b),
dset_set: dset_set,
dset_list: s List,
dset_of_mon: g↓set,
top: Top,
set_eq: =b,
pi2: snd(t),
and: P ∧ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
not: ¬A,
or: P ∨ Q,
false: False,
infix_ap: x f y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
not_wf,
equal_wf,
grp_car_wf,
grp_id_wf,
assert_wf,
before_wf,
map_wf,
set_car_wf,
set_prod_wf,
dset_of_mon_wf,
oalist_wf,
dset_wf,
abdmonoid_wf,
loset_wf,
cons_wf,
map_cons_lemma,
sd_ordered_cons_lemma,
mem_cons_lemma,
assert_of_band,
sd_ordered_wf,
mem_wf,
dset_of_mon_wf0,
or_wf,
bor_wf,
infix_ap_wf,
bool_wf,
grp_eq_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_mon_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
because_Cache,
dependent_functionElimination,
applyEquality,
sqequalRule,
lambdaEquality,
productElimination,
dependent_set_memberEquality,
productEquality,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_pairFormation,
unionElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
addLevel,
impliesFunctionality,
orFunctionality
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}ws:|oal(a;b)|. \mforall{}x:|a|. \mforall{}y:|b|.
((\muparrow{}before(x;map(\mlambda{}x.(fst(x));ws))) {}\mRightarrow{} (\mneg{}(y = e)) {}\mRightarrow{} ([<x, y> / ws] \mmember{} |oal(a;b)|))
Date html generated:
2017_10_01-AM-10_01_45
Last ObjectModification:
2017_03_03-PM-01_04_03
Theory : polynom_2
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