Nuprl Lemma : lookup_before_start_a
∀a:QOSet. ∀b:AbMon. ∀k:|a|. ∀ps:(|a| × |b|) List.
((↑(∀bk'(:|a|) ∈ map(λz.(fst(z));ps). (k' <b k))) ⇒ ((ps[k]) = e ∈ |b|))
Proof
Definitions occuring in Statement :
lookup: as[k],
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],
equal: s = t ∈ T,
abmonoid: AbMon,
grp_id: e,
grp_car: |g|,
qoset: QOSet,
set_blt: a <b b,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
qoset: QOSet,
dset: DSet,
abmonoid: AbMon,
mon: Mon,
so_lambda: λ2x.t[x],
prop: ℙ,
pi1: fst(t),
so_apply: x[s],
ball: ball,
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
band: p ∧b q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y,
not: ¬A,
false: False,
guard: {T}
Lemmas referenced :
list_induction,
set_car_wf,
grp_car_wf,
assert_wf,
ball_wf,
map_wf,
set_blt_wf,
equal_wf,
lookup_wf,
grp_id_wf,
list_wf,
pi1_wf,
abmonoid_wf,
qoset_wf,
map_nil_lemma,
lookup_nil_lemma,
ball_nil_lemma,
true_wf,
map_cons_lemma,
lookup_cons_pr_lemma,
ball_cons_lemma,
iff_transitivity,
bool_wf,
eqtt_to_assert,
assert_of_set_lt,
set_lt_wf,
iff_weakening_uiff,
assert_of_band,
set_eq_wf,
uiff_transitivity,
equal-wf-T-base,
assert_of_dset_eq,
bnot_wf,
not_wf,
eqff_to_assert,
assert_of_bnot,
set_lt_transitivity_2,
set_leq_weakening_eq,
set_lt_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
productEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
functionEquality,
dependent_functionElimination,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
applyEquality,
baseClosed,
impliesFunctionality
Latex:
\mforall{}a:QOSet. \mforall{}b:AbMon. \mforall{}k:|a|. \mforall{}ps:(|a| \mtimes{} |b|) List.
((\muparrow{}(\mforall{}\msubb{}k'(:|a|) \mmember{} map(\mlambda{}z.(fst(z));ps). (k' <\msubb{} k))) {}\mRightarrow{} ((ps[k]) = e))
Date html generated:
2017_10_01-AM-10_02_17
Last ObjectModification:
2017_03_03-PM-01_04_38
Theory : polynom_2
Home
Index