Nuprl Lemma : lookup_oal_cons
∀a:LOSet. ∀b:OCMon. ∀k,kp:|a|. ∀vp:|b|. ∀ps:|oal(a;b)|.
((↑before(kp;map(λz.(fst(z));ps))) ⇒ (([<kp, vp> / ps][k]) = ((when kp (=b) k. vp) * (ps[k])) ∈ |b|))
Proof
Definitions occuring in Statement :
lookup: as[k],
oalist: oal(a;b),
before: before(u;ps),
map: map(f;as),
cons: [a / b],
assert: ↑b,
infix_ap: x f y,
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
pair: <a, b>,
equal: s = t ∈ T,
mon_when: when b. p,
ocmon: OCMon,
grp_id: e,
grp_op: *,
grp_car: |g|,
loset: LOSet,
set_eq: =b,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
dset: DSet,
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,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
mon_when: when b. p,
top: Top,
infix_ap: x f y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
squash: ↓T,
true: True
Lemmas referenced :
assert_wf,
before_wf,
map_wf,
set_car_wf,
set_prod_wf,
dset_of_mon_wf,
abdmonoid_dmon,
ocmon_subtype_abdmonoid,
subtype_rel_transitivity,
ocmon_wf,
abdmonoid_wf,
dmon_wf,
oalist_wf,
dset_wf,
grp_car_wf,
loset_wf,
lookup_cons_pr_lemma,
set_eq_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
equal_wf,
eqtt_to_assert,
assert_of_dset_eq,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
squash_wf,
true_wf,
grp_op_wf,
lookup_wf,
list_wf,
poset_sig_wf,
grp_id_wf,
iff_weakening_equal,
lookup_before_start,
mon_ident,
iabmonoid_subtype_imon,
abmonoid_subtype_iabmonoid,
abdmonoid_abmonoid,
abmonoid_wf,
iabmonoid_wf,
imon_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
setElimination,
rename,
because_Cache,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
lambdaEquality,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
independent_functionElimination,
independent_pairFormation,
impliesFunctionality,
imageElimination,
universeEquality,
productEquality,
cumulativity,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}a:LOSet. \mforall{}b:OCMon. \mforall{}k,kp:|a|. \mforall{}vp:|b|. \mforall{}ps:|oal(a;b)|.
((\muparrow{}before(kp;map(\mlambda{}z.(fst(z));ps))) {}\mRightarrow{} (([<kp, vp> / ps][k]) = ((when kp (=\msubb{}) k. vp) * (ps[k]))))
Date html generated:
2017_10_01-AM-10_02_15
Last ObjectModification:
2017_03_03-PM-01_04_35
Theory : polynom_2
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