Nuprl Lemma : before_all_imp_before
∀a:LOSet. ∀b:AbMon. ∀k:|a|. ∀ps:(|a| × |b|) List.
((↑(∀bx(:|a|) ∈ map(λz.(fst(z));ps). (x <b k))) ⇒ (↑before(k;map(λz.(fst(z));ps))))
Proof
Definitions occuring in Statement :
before: before(u;ps),
ball: ball,
map: map(f;as),
list: T List,
assert: ↑b,
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
product: x:A × B[x],
abmonoid: AbMon,
grp_car: |g|,
loset: LOSet,
set_blt: a <b b,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
abmonoid: AbMon,
mon: Mon,
or: P ∨ Q,
cons: [a / b],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
implies: P ⇒ Q,
true: True,
prop: ℙ,
and: P ∧ Q,
pi1: fst(t),
ball: ball,
bool: 𝔹,
unit: Unit,
it: ⋅,
band: p ∧b q,
uiff: uiff(P;Q),
uimplies: b supposing a,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
set_car_wf,
grp_car_wf,
list-cases,
product_subtype_list,
list_wf,
abmonoid_wf,
loset_wf,
map_nil_lemma,
ball_nil_lemma,
before_nil_lemma,
true_wf,
map_cons_lemma,
ball_cons_lemma,
before_cons_lemma,
set_lt_wf,
assert_wf,
ball_wf,
map_wf,
set_blt_wf,
iff_transitivity,
bool_wf,
eqtt_to_assert,
assert_of_set_lt,
equal_wf,
iff_weakening_uiff,
assert_of_band,
pi1_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
productEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
because_Cache,
lambdaEquality,
addLevel,
impliesFunctionality,
equalityElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
independent_pairEquality,
independent_pairFormation,
functionEquality
Latex:
\mforall{}a:LOSet. \mforall{}b:AbMon. \mforall{}k:|a|. \mforall{}ps:(|a| \mtimes{} |b|) List.
((\muparrow{}(\mforall{}\msubb{}x(:|a|) \mmember{} map(\mlambda{}z.(fst(z));ps). (x <\msubb{} k))) {}\mRightarrow{} (\muparrow{}before(k;map(\mlambda{}z.(fst(z));ps))))
Date html generated:
2017_10_01-AM-10_01_43
Last ObjectModification:
2017_03_03-PM-01_03_57
Theory : polynom_2
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