Nuprl Lemma : oal_lk_merge_2
∀s:LOSet. ∀g:AbDMon. ∀ps,qs:|oal(s;g)|.
((¬(ps = 00 ∈ |oal(s;g)|))
⇒ (¬(qs = 00 ∈ |oal(s;g)|))
⇒ (¬((ps ++ qs) = 00 ∈ |oal(s;g)|))
⇒ (lk(ps) = lk(qs) ∈ |s|)
⇒ (¬((lv(ps) * lv(qs)) = e ∈ |g|))
⇒ (lk(ps ++ qs) = lk(qs) ∈ |s|))
Proof
Definitions occuring in Statement :
oal_lv: lv(ps),
oal_lk: lk(ps),
oal_merge: ps ++ qs,
oal_nil: 00,
oalist: oal(a;b),
infix_ap: x f y,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
equal: s = t ∈ T,
abdmonoid: AbDMon,
grp_id: e,
grp_op: *,
grp_car: |g|,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
prop: ℙ,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
infix_ap: x f y,
so_apply: x[s],
guard: {T},
nil: [],
it: ⋅,
oal_nil: 00,
squash: ↓T,
true: True,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
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,
respects-equality: respects-equality(S;T),
oal_cons_pr: oal_cons_pr(x;y;ws),
oal_lv: lv(ps),
top: Top,
pi2: snd(t),
oal_lk: lk(ps),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
oalist_cases_a,
all_wf,
set_car_wf,
oalist_wf,
not_wf,
equal_wf,
oal_nil_wf,
oal_merge_wf2,
oal_lk_wf,
grp_car_wf,
infix_ap_wf,
grp_op_wf,
oal_lv_wf,
grp_id_wf,
abdmonoid_wf,
loset_wf,
squash_wf,
true_wf,
istype-universe,
oal_nil_ident_l,
subtype_rel_self,
iff_weakening_equal,
istype-void,
oal_merge_wf,
nil_wf,
subtype-respects-equality,
list_wf,
oal_cons_pr_wf,
oal_nil_ident_r,
istype-assert,
before_wf,
map_wf,
set_prod_wf,
dset_of_mon_wf,
reduce_hd_cons_lemma,
oal_merge_conses_lemma,
set_blt_wf,
eqtt_to_assert,
assert_of_set_lt,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
set_lt_wf,
grp_eq_wf,
assert_of_mon_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
isectElimination,
hypothesis,
applyEquality,
because_Cache,
functionEquality,
setElimination,
rename,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
universeIsType,
imageElimination,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
natural_numberEquality,
independent_isectElimination,
productElimination,
functionIsType,
equalityIstype,
productEquality,
voidElimination,
isect_memberEquality_alt,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
promote_hyp,
cumulativity
Latex:
\mforall{}s:LOSet. \mforall{}g:AbDMon. \mforall{}ps,qs:|oal(s;g)|.
((\mneg{}(ps = 00))
{}\mRightarrow{} (\mneg{}(qs = 00))
{}\mRightarrow{} (\mneg{}((ps ++ qs) = 00))
{}\mRightarrow{} (lk(ps) = lk(qs))
{}\mRightarrow{} (\mneg{}((lv(ps) * lv(qs)) = e))
{}\mRightarrow{} (lk(ps ++ qs) = lk(qs)))
Date html generated:
2019_10_16-PM-01_08_10
Last ObjectModification:
2018_11_27-AM-11_32_20
Theory : polynom_2
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