Nuprl Lemma : oal_dom_inj
∀a:LOSet. ∀b:AbDMon. ∀k:|a|. ∀v:|b|. (dom(inj(k,v)) = if v =b e then 0{a} else mset_inj{a}(k) fi ∈ FiniteSet{a})
Proof
Definitions occuring in Statement :
oal_inj: inj(k,v),
oal_dom: dom(ps),
mset_inj: mset_inj{s}(x),
finite_set: FiniteSet{s},
null_mset: 0{s},
ifthenelse: if b then t else f fi ,
infix_ap: x f y,
all: ∀x:A. B[x],
equal: s = t ∈ T,
abdmonoid: AbDMon,
grp_id: e,
grp_eq: =b,
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],
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
finite_set: FiniteSet{s},
oal_inj: inj(k,v),
infix_ap: x f y,
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 ,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
prop: ℙ,
rev_implies: P ⇐ Q,
false: False,
guard: {T},
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
null_mset: 0{s},
oal_dom: dom(ps),
mk_mset: mk_mset(as),
top: Top,
mset_inj: mset_inj{s}(x),
mset_sum: a + b,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
pi1: fst(t),
squash: ↓T,
true: True
Lemmas referenced :
grp_car_wf,
set_car_wf,
abdmonoid_wf,
loset_wf,
oal_dom_wf2,
oal_inj_wf,
grp_eq_wf,
grp_id_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
equal_wf,
eqtt_to_assert,
assert_of_mon_eq,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
fset_properties,
all_wf,
le_wf,
mset_count_wf,
nat_wf,
map_nil_lemma,
null_mset_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
map_cons_lemma,
squash_wf,
true_wf,
mset_wf,
mset_sum_ident_r,
mset_inj_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
dependent_functionElimination,
dependent_set_memberEquality,
applyEquality,
because_Cache,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
independent_functionElimination,
productElimination,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
impliesFunctionality,
voidElimination,
lambdaEquality,
natural_numberEquality,
isect_memberEquality,
voidEquality,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}k:|a|. \mforall{}v:|b|. (dom(inj(k,v)) = if v =\msubb{} e then 0\{a\} else mset\_inj\{a\}(k) fi )
Date html generated:
2017_10_01-AM-10_03_03
Last ObjectModification:
2017_03_03-PM-01_05_35
Theory : polynom_2
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