Nuprl Lemma : lookup_oal_inj
∀a:LOSet. ∀b:AbDMon. ∀k,k':|a|. ∀v:|b|. ((inj(k,v)[k']) = (when k (=b) k'. v) ∈ |b|)
Proof
Definitions occuring in Statement :
oal_inj: inj(k,v),
lookup: as[k],
infix_ap: x f y,
all: ∀x:A. B[x],
equal: s = t ∈ T,
mon_when: when b. p,
abdmonoid: AbDMon,
grp_id: e,
grp_car: |g|,
loset: LOSet,
set_eq: =b,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
oal_inj: inj(k,v),
member: t ∈ T,
infix_ap: x f y,
uall: ∀[x:A]. B[x],
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
top: Top,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
prop: ℙ,
rev_implies: P ⇐ Q,
false: False,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
mon_when: when b. p,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
grp_eq_wf,
grp_id_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
equal_wf,
grp_car_wf,
eqtt_to_assert,
assert_of_mon_eq,
lookup_nil_lemma,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
lookup_cons_pr_lemma,
set_car_wf,
abdmonoid_wf,
loset_wf,
set_eq_wf,
assert_of_dset_eq,
mon_when_wf,
iabmonoid_subtype_imon,
abmonoid_subtype_iabmonoid,
abdmonoid_abmonoid,
subtype_rel_transitivity,
abmonoid_wf,
iabmonoid_wf,
imon_wf,
infix_ap_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
independent_functionElimination,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
impliesFunctionality,
instantiate
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}k,k':|a|. \mforall{}v:|b|. ((inj(k,v)[k']) = (when k (=\msubb{}) k'. v))
Date html generated:
2017_10_01-AM-10_02_59
Last ObjectModification:
2017_03_03-PM-01_05_29
Theory : polynom_2
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