Nuprl Lemma : oal_bpos_trichot
∀s:LOSet. ∀g:OGrp. ∀rs:|oal(s;g)|. ((↑pos(rs)) ∨ (rs = 00 ∈ |oal(s;g)|) ∨ (↑pos(--rs)))
Proof
Definitions occuring in Statement :
oal_bpos: pos(ps),
oal_neg: --ps,
oal_nil: 00,
oalist: oal(a;b),
assert: ↑b,
all: ∀x:A. B[x],
or: P ∨ Q,
equal: s = t ∈ T,
ocgrp: OGrp,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
dset: DSet,
oal_bpos: pos(ps),
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,
band: p ∧b q,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
assert: ↑b,
or: P ∨ Q,
list: T List,
grp_car: |g|,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
ocgrp: OGrp,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
prop: ℙ,
pi2: snd(t),
false: False,
exists: ∃x:A. B[x],
sq_type: SQType(T),
not: ¬A,
rev_implies: P ⇐ Q,
respects-equality: respects-equality(S;T),
squash: ↓T,
true: True
Lemmas referenced :
omon_inc,
ocmon_subtype_omon,
ocgrp_subtype_ocmon,
subtype_rel_transitivity,
ocgrp_wf,
ocmon_wf,
omon_wf,
ocgrp_abdgrp,
set_car_wf,
oalist_wf,
loset_wf,
oal_neg_wf2,
oal_null_wf,
eqtt_to_assert,
assert_of_oal_null,
subtype_rel_self,
list_wf,
grp_car_wf,
mon_subtype_grp_sig,
dmon_subtype_mon,
abdmonoid_dmon,
ocmon_subtype_abdmonoid,
abdmonoid_wf,
dmon_wf,
mon_wf,
grp_sig_wf,
assert_wf,
sd_ordered_wf,
map_wf,
not_wf,
mem_wf,
dset_of_mon_wf,
grp_id_wf,
dset_of_mon_wf0,
equal_functionality_wrt_subtype_rel2,
istype-void,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
istype-assert,
grp_blt_wf,
oal_lv_wf,
oal_nil_wf,
oal_neg_eq_nil,
iff_weakening_uiff,
grp_lt_wf,
assert_of_grp_blt,
subtype-respects-equality,
grp_inv_wf,
uiff_transitivity2,
squash_wf,
true_wf,
oal_lv_neg,
grp_lt_trichot,
oal_lv_nid,
grp_lt_shift_right,
mon_ident,
iabmonoid_subtype_imon,
abmonoid_subtype_iabmonoid,
abdmonoid_abmonoid,
abmonoid_wf,
iabmonoid_wf,
imon_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
universeIsType,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
unionElimination,
equalityElimination,
productElimination,
independent_functionElimination,
because_Cache,
inrFormation_alt,
inlFormation_alt,
setEquality,
productEquality,
productIsType,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
cumulativity,
voidElimination,
unionIsType,
setIsType,
functionIsType,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}s:LOSet. \mforall{}g:OGrp. \mforall{}rs:|oal(s;g)|. ((\muparrow{}pos(rs)) \mvee{} (rs = 00) \mvee{} (\muparrow{}pos(--rs)))
Date html generated:
2019_10_16-PM-01_08_32
Last ObjectModification:
2018_11_27-AM-10_31_02
Theory : polynom_2
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