Nuprl Lemma : lookup_before_start
∀a:LOSet. ∀b:AbDMon. ∀k:|a|. ∀ps:|oal(a;b)|. ((↑before(k;map(λz.(fst(z));ps))) ⇒ ((ps[k]) = e ∈ |b|))
Proof
Definitions occuring in Statement :
lookup: as[k],
oalist: oal(a;b),
before: before(u;ps),
map: map(f;as),
assert: ↑b,
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
equal: s = t ∈ T,
abdmonoid: AbDMon,
grp_id: e,
grp_car: |g|,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
dset: DSet,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
abdmonoid: AbDMon,
set_prod: s × t,
mk_dset: mk_dset(T, eq),
set_car: |p|,
pi1: fst(t),
oalist: oal(a;b),
dset_set: dset_set,
dset_list: s List,
dset_of_mon: g↓set,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
dmon: DMon,
mon: Mon,
so_apply: x[s],
map: map(f;as),
list_ind: list_ind,
nil: [],
it: ⋅,
infix_ap: x f y,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
before: before(u;ps),
bor: p ∨bq
Lemmas referenced :
set_car_wf,
oalist_wf,
dset_wf,
abdmonoid_wf,
loset_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
assert_wf,
before_wf,
map_wf,
set_prod_wf,
dset_of_mon_wf,
le_wf,
length_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
non_neg_length,
decidable__lt,
lelt_wf,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
oalist_cases,
all_wf,
grp_car_wf,
equal_wf,
lookup_wf,
grp_id_wf,
list_wf,
lookup_nil_lemma,
nil_wf,
lookup_cons_pr_lemma,
cons_wf,
not_wf,
set_eq_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
eqtt_to_assert,
assert_of_dset_eq,
iff_transitivity,
bnot_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
map_cons_lemma,
null_cons_lemma,
reduce_hd_cons_lemma,
assert_of_set_lt,
qoset_lt_irrefl,
length_of_cons_lemma,
before_cons_lemma,
before_trans,
length_wf_nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
because_Cache,
productElimination,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
imageElimination,
addEquality,
functionEquality,
productEquality,
independent_pairEquality,
equalityElimination,
baseClosed,
impliesFunctionality
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{} ((ps[k]) = e))
Date html generated:
2017_10_01-AM-10_02_12
Last ObjectModification:
2017_03_03-PM-01_05_01
Theory : polynom_2
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