Nuprl Lemma : oal_lk_bounds_dom
∀s:LOSet. ∀g:AbDMon. ∀k:|s|. ∀ps:|oal(s;g)|. ((¬(ps = 00 ∈ |oal(s;g)|)) ⇒ (↑(k ∈b dom(ps))) ⇒ (k ≤ lk(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_leq: a ≤ b,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
dset: DSet,
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,
oal_nil: 00,
nil: [],
it: ⋅,
respects-equality: respects-equality(S;T),
mon: Mon,
dmon: DMon,
abdmonoid: AbDMon,
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_lambda3,
top: Top,
so_apply: x[s1;s2;s3],
oal_lk: lk(ps),
mset_mem: mset_mem,
iff: P ⇐⇒ Q,
and: P ∧ Q,
infix_ap: x f y,
or: P ∨ Q,
rev_implies: P ⇐ Q,
uimplies: b supposing a
Lemmas referenced :
oalist_cases_b,
not_wf,
equal_wf,
set_car_wf,
oalist_wf,
oal_nil_wf,
assert_wf,
mset_mem_wf,
oal_dom_wf,
abdmonoid_abmonoid,
set_leq_wf,
oal_lk_wf,
abdmonoid_wf,
loset_wf,
istype-void,
void-list-equality,
nil_wf,
set_blt_wf,
map_wf,
ball_wf,
grp_id_wf,
respects-equality-oalist1,
dset_of_mon_wf,
set_prod_wf,
grp_car_wf,
cons_wf,
istype-assert,
list_ind_cons_lemma,
list_ind_nil_lemma,
map_cons_lemma,
mset_mem_inj_sum_lemma,
reduce_hd_cons_lemma,
ball_char,
iff_weakening_uiff,
set_lt_wf,
assert_of_set_lt,
mem_wf,
iff_transitivity,
bor_wf,
infix_ap_wf,
bool_wf,
set_eq_wf,
or_wf,
assert_of_bor,
assert_of_dset_eq,
set_leq_weakening_eq,
set_leq_weakening_lt
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,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
universeIsType,
because_Cache,
voidElimination,
functionIsType,
equalityIstype,
baseClosed,
sqequalBase,
voidEquality,
productElimination,
independent_pairEquality,
productEquality,
isect_memberEquality_alt,
independent_pairFormation,
unionElimination,
inlFormation_alt,
inrFormation_alt,
equalityIsType1,
unionIsType,
independent_isectElimination
Latex:
\mforall{}s:LOSet. \mforall{}g:AbDMon. \mforall{}k:|s|. \mforall{}ps:|oal(s;g)|. ((\mneg{}(ps = 00)) {}\mRightarrow{} (\muparrow{}(k \mmember{}\msubb{} dom(ps))) {}\mRightarrow{} (k \mleq{} lk(ps)))
Date html generated:
2020_05_20-AM-09_36_09
Last ObjectModification:
2020_01_08-PM-06_00_14
Theory : polynom_2
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