Nuprl Lemma : before_imp_before_all
∀a:LOSet. ∀b:AbDMon. ∀k:|a|. ∀ps:|oal(a;b)|.
((↑before(k;map(λz.(fst(z));ps))) ⇒ (↑(∀bx(:|a|) ∈ map(λz.(fst(z));ps). (x <b k))))
Proof
Definitions occuring in Statement :
oalist: oal(a;b),
before: before(u;ps),
ball: ball,
map: map(f;as),
assert: ↑b,
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
abdmonoid: AbDMon,
loset: LOSet,
set_blt: a <b b,
set_car: |p|
Definitions unfolded in proof :
dset_of_mon: g↓set,
dset_list: s List,
dset_set: dset_set,
oalist: oal(a;b),
pi1: fst(t),
set_car: |p|,
mk_dset: mk_dset(T, eq),
set_prod: s × t,
dset: DSet,
subtype_rel: A ⊆r B,
abdmonoid: AbDMon,
qoset: QOSet,
poset: POSet{i},
loset: LOSet,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
squash: ↓T,
sq_stable: SqStable(P),
so_apply: x[s],
so_lambda: λ2x.t[x],
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
top: Top,
infix_ap: x f y,
pi2: snd(t),
grp_op: *,
guard: {T},
so_apply: x[s1;s2],
grp_car: |g|,
band_mon: <𝔹,∧b>,
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
assert_functionality_wrt_uiff,
cons_wf,
mon_htfor_wf,
band_mon_wf,
bool_wf,
list_wf,
sd_ordered_char,
mon_htfor_cons_lemma,
sd_ordered_cons_lemma,
assert_of_band,
sd_ordered_wf,
sq_stable_from_decidable,
ball_wf,
set_blt_wf,
decidable__assert,
assert_wf,
before_wf,
map_wf,
set_car_wf,
set_prod_wf,
dset_of_mon_wf,
oalist_wf,
dset_wf,
abdmonoid_wf,
loset_wf
Rules used in proof :
productElimination,
lambdaEquality,
sqequalRule,
applyEquality,
because_Cache,
hypothesis,
hypothesisEquality,
rename,
setElimination,
dependent_functionElimination,
thin,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageElimination,
baseClosed,
imageMemberEquality,
introduction,
independent_functionElimination,
independent_pairFormation,
independent_isectElimination,
voidEquality,
voidElimination,
isect_memberEquality
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}k:|a|. \mforall{}ps:|oal(a;b)|.
((\muparrow{}before(k;map(\mlambda{}z.(fst(z));ps))) {}\mRightarrow{} (\muparrow{}(\mforall{}\msubb{}x(:|a|) \mmember{} map(\mlambda{}z.(fst(z));ps). (x <\msubb{} k))))
Date html generated:
2016_05_16-AM-08_15_35
Last ObjectModification:
2016_01_16-PM-11_58_34
Theory : polynom_2
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