Nuprl Lemma : assert_of_oal_blt
∀s:LOSet. ∀g:OGrp. ∀ps,qs:|oal(s;g)|. (↑(ps <<b qs) ⇐⇒ ps << qs)
Proof
Definitions occuring in Statement :
oal_lt: ps << qs,
oal_blt: ps <<b qs,
oalist: oal(a;b),
assert: ↑b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
ocgrp: OGrp,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
oal_lt: ps << qs,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
dset: DSet,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
ocgrp: OGrp,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
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,
infix_ap: x f y,
prop: ℙ,
grp_car: |g|,
oal_blt: ps <<b qs,
oal_bpos: pos(ps),
band: p ∧b q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
not: ¬A,
false: False,
ifthenelse: if b then t else f fi ,
bfalse: ff,
assert: ↑b,
squash: ↓T,
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
or: P ∨ Q,
oal_nil: 00,
nil: [],
top: Top
Lemmas referenced :
set_car_wf,
oalist_wf,
ocmon_subtype_abdmonoid,
ocgrp_subtype_ocmon,
subtype_rel_transitivity,
ocgrp_wf,
ocmon_wf,
abdmonoid_wf,
loset_wf,
iff_weakening_uiff,
equal_wf,
grp_car_wf,
lookup_wf,
grp_id_wf,
grp_op_wf,
grp_inv_wf,
grp_eq_shift_right,
grp_subtype_igrp,
abgrp_subtype_grp,
ocgrp_subtype_abgrp,
abgrp_wf,
grp_wf,
igrp_wf,
set_lt_wf,
grp_lt_wf,
grp_lt_shift_right,
dset_of_mon_wf0,
subtype_rel_self,
istype-assert,
oal_blt_wf,
ocgrp_subtype_abdgrp,
oal_neg_wf2,
oal_merge_wf2,
bnot_wf,
oal_null_wf,
iff_transitivity,
equal-wf-T-base,
bool_wf,
assert_wf,
not_wf,
oal_nil_wf,
eqtt_to_assert,
assert_of_bnot,
assert_of_oal_null,
istype-void,
eqff_to_assert,
grp_blt_wf,
oal_lv_wf,
oal_lk_wf,
squash_wf,
true_wf,
istype-universe,
lookup_oal_neg,
lookup_merge,
iff_weakening_equal,
lookup_oal_eq_id,
oal_lk_bounds_dom,
set_lt_transitivity_2,
set_lt_irreflexivity,
mset_mem_wf,
oal_dom_wf,
oal_merge_wf,
oal_neg_wf,
grp_sig_wf,
lookup_oal_lk,
assert_of_grp_blt,
loset_trichot,
oal_lv_nid,
list_wf,
poset_sig_wf,
grp_lt_irreflexivity,
dneg_elim,
decidable__dset_eq,
oal_equal_char,
nil_wf,
lookup_nil_lemma,
ocmon_subtype_omon,
omon_wf,
grp_lt_transitivity_2,
grp_leq_weakening_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
inhabitedIsType,
hypothesisEquality,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination,
dependent_pairFormation_alt,
because_Cache,
independent_pairFormation,
promote_hyp,
productIsType,
functionIsType,
equalityIstype,
unionElimination,
equalityElimination,
baseClosed,
voidElimination,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
productEquality,
isect_memberEquality_alt
Latex:
\mforall{}s:LOSet. \mforall{}g:OGrp. \mforall{}ps,qs:|oal(s;g)|. (\muparrow{}(ps <<\msubb{} qs) \mLeftarrow{}{}\mRightarrow{} ps << qs)
Date html generated:
2019_10_16-PM-01_08_26
Last ObjectModification:
2018_11_27-AM-10_42_25
Theory : polynom_2
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