Nuprl Lemma : oal_lk_in_dom
∀s:LOSet. ∀g:AbDMon. ∀ps:|oal(s;g)|. ((¬(ps = 00 ∈ |oal(s;g)|)) ⇒ (↑(lk(ps) ∈b dom(ps))))
Proof
Definitions occuring in Statement :
oal_lk: lk(ps),
oal_dom: dom(ps),
oal_nil: 00,
oalist: oal(a;b),
mset_mem: mset_mem,
assert: ↑b,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
equal: s = t ∈ T,
abdmonoid: AbDMon,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
set_car: |p|,
pi1: fst(t),
dset_list: s List,
set_prod: s × t,
dset_of_mon: g↓set,
so_apply: x[s],
guard: {T},
not: ¬A,
false: False,
dset: DSet,
oal_nil: 00,
and: P ∧ Q,
cand: A c∧ B,
assert: ↑b,
ifthenelse: if b then t else f fi ,
sd_ordered: sd_ordered(as),
ycomb: Y,
list_ind: list_ind,
map: map(f;as),
nil: [],
it: ⋅,
btrue: tt,
true: True,
mem: a ∈b as,
mon_for: For{g} x ∈ as. f[x],
for: For{T,op,id} x ∈ as. f[x],
reduce: reduce(f;k;as),
grp_id: e,
pi2: snd(t),
bor_mon: <𝔹,∨b>,
bfalse: ff,
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
grp_car: |g|,
oal_lk: lk(ps),
top: Top,
mset_inj: mset_inj{s}(x),
oal_dom: dom(ps),
mset_sum: a + b,
mk_mset: mk_mset(as),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
infix_ap: x f y,
iff: P ⇐⇒ Q,
or: P ∨ Q,
rev_implies: P ⇐ Q,
list: T List
Lemmas referenced :
oalist_cases_a,
not_wf,
equal_wf,
set_car_wf,
oalist_wf,
oal_nil_wf,
assert_wf,
mset_mem_wf,
oal_lk_wf,
oal_dom_wf,
abdmonoid_abmonoid,
abdmonoid_wf,
loset_wf,
equal-wf-base-T,
mem_wf,
dset_of_mon_wf,
grp_id_wf,
subtype_rel_self,
dset_of_mon_wf0,
nil_wf,
grp_car_wf,
sd_ordered_wf,
map_wf,
cons_pr_in_oalist,
before_wf,
set_prod_wf,
reduce_hd_cons_lemma,
istype-void,
list_ind_cons_lemma,
list_ind_nil_lemma,
map_cons_lemma,
mset_mem_inj_sum_lemma,
iff_transitivity,
bor_wf,
set_eq_wf,
or_wf,
member_wf,
iff_weakening_uiff,
assert_of_bor,
assert_of_dset_eq,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
functionEquality,
isectElimination,
hypothesis,
applyEquality,
because_Cache,
setElimination,
rename,
independent_functionElimination,
universeIsType,
voidElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
baseClosed,
natural_numberEquality,
independent_pairFormation,
dependent_set_memberEquality_alt,
productEquality,
productIsType,
productElimination,
isect_memberEquality_alt,
unionElimination,
inlFormation_alt,
inrFormation_alt,
equalityIsType1,
unionIsType,
setEquality
Latex:
\mforall{}s:LOSet. \mforall{}g:AbDMon. \mforall{}ps:|oal(s;g)|. ((\mneg{}(ps = 00)) {}\mRightarrow{} (\muparrow{}(lk(ps) \mmember{}\msubb{} dom(ps))))
Date html generated:
2019_10_16-PM-01_07_53
Last ObjectModification:
2018_10_08-PM-06_38_12
Theory : polynom_2
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