Nuprl Lemma : oal_lt_trichot
∀s:LOSet. ∀g:OGrp. ∀ps,qs:|oal(s;g)|. ((ps << qs) ∨ (ps = qs ∈ |oal(s;g)|) ∨ (qs << ps))
Proof
Definitions occuring in Statement :
oal_lt: ps << qs,
oalist: oal(a;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,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
ocgrp: OGrp,
prop: ℙ,
dset: DSet,
or: P ∨ Q,
oal_blt: ps <<b qs,
oal_grp: oal_grp(s;g),
grp_car: |g|,
pi1: fst(t),
grp_op: *,
pi2: snd(t),
infix_ap: x f y,
grp_inv: ~,
grp_id: e,
squash: ↓T,
abdgrp: AbDGrp,
abgrp: AbGrp,
grp: Group{i},
mon: Mon,
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
set_car: |p|,
dset_list: s List,
set_prod: s × t,
dset_of_mon: g↓set,
true: True,
imon: IMonoid
Lemmas referenced :
ocgrp_wf,
loset_wf,
ocgrp_subtype_abdgrp,
ocmon_subtype_abdmonoid,
ocgrp_subtype_ocmon,
subtype_rel_transitivity,
ocmon_wf,
abdmonoid_wf,
assert_of_oal_blt,
oal_lt_wf,
or_wf,
equal_wf,
set_car_wf,
oalist_wf,
dset_wf,
assert_wf,
oal_blt_wf,
oal_bpos_trichot,
oal_merge_wf2,
oal_neg_wf2,
oal_bpos_wf,
squash_wf,
true_wf,
grp_car_wf,
oal_grp_wf,
abdgrp_wf,
grp_op_wf,
infix_ap_wf,
iff_weakening_equal,
grp_sig_wf,
monoid_p_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
comm_wf,
eqfun_p_wf,
grp_eq_wf,
grp_subtype_igrp,
abgrp_subtype_grp,
abdgrp_subtype_abgrp,
abgrp_wf,
grp_wf,
igrp_wf,
mon_assoc,
grp_inverse,
mon_ident,
grp_inv_thru_op,
grp_inv_inv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
hypothesisEquality,
applyEquality,
sqequalHypSubstitution,
sqequalRule,
independent_pairFormation,
thin,
instantiate,
isectElimination,
independent_isectElimination,
productElimination,
addLevel,
orFunctionality,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
lambdaEquality,
because_Cache,
unionElimination,
inlFormation,
inrFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
functionEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
setEquality,
cumulativity,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}s:LOSet. \mforall{}g:OGrp. \mforall{}ps,qs:|oal(s;g)|. ((ps << qs) \mvee{} (ps = qs) \mvee{} (qs << ps))
Date html generated:
2017_10_01-AM-10_04_19
Last ObjectModification:
2017_03_03-PM-01_07_25
Theory : polynom_2
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