Nuprl Lemma : oal_merge_preserves_lt
∀s:LOSet. ∀g:OCMon. ∀ps,qs,rs:|oal(s;g)|. ((qs << rs) ⇒ ((ps ++ qs) << (ps ++ rs)))
Proof
Definitions occuring in Statement :
oal_lt: ps << qs,
oal_merge: ps ++ qs,
oalist: oal(a;b),
all: ∀x:A. B[x],
implies: P ⇒ Q,
ocmon: OCMon,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
oal_lt: ps << qs,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
dset: DSet,
squash: ↓T,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
so_lambda: λ2x.t[x],
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],
infix_ap: x f y
Lemmas referenced :
oal_lt_wf,
set_car_wf,
oalist_wf,
ocmon_subtype_abdmonoid,
dset_wf,
ocmon_wf,
loset_wf,
equal_wf,
squash_wf,
true_wf,
grp_car_wf,
lookup_merge,
iff_weakening_equal,
set_lt_wf,
grp_lt_wf,
grp_sig_wf,
all_wf,
lookup_wf,
grp_id_wf,
oal_merge_wf,
grp_op_wf,
infix_ap_wf,
dset_of_mon_wf0,
grp_op_preserves_lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
isectElimination,
applyEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
because_Cache,
dependent_pairFormation,
independent_pairFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
productEquality,
functionEquality
Latex:
\mforall{}s:LOSet. \mforall{}g:OCMon. \mforall{}ps,qs,rs:|oal(s;g)|. ((qs << rs) {}\mRightarrow{} ((ps ++ qs) << (ps ++ rs)))
Date html generated:
2017_10_01-AM-10_04_22
Last ObjectModification:
2017_03_03-PM-01_07_45
Theory : polynom_2
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