Nuprl Lemma : oal_inj_wf
∀a:LOSet. ∀b:AbDMon. ∀k:|a|. ∀v:|b|. (inj(k,v) ∈ |oal(a;b)|)
Proof
Definitions occuring in Statement :
oal_inj: inj(k,v),
oalist: oal(a;b),
all: ∀x:A. B[x],
member: t ∈ T,
abdmonoid: AbDMon,
grp_car: |g|,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
oal_inj: inj(k,v),
all: ∀x:A. B[x],
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 ,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
set_car: |p|,
pi1: fst(t),
dset_list: s List,
set_prod: s × t,
dset_of_mon: g↓set,
top: Top,
set_eq: =b,
pi2: snd(t),
band: p ∧b q,
assert: ↑b,
cand: A c∧ B,
true: True,
subtype_rel: A ⊆r B,
grp_car: |g|,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
prop: ℙ,
guard: {T}
Lemmas referenced :
grp_eq_wf,
grp_id_wf,
uiff_transitivity,
equal-wf-T-base,
bool_wf,
assert_wf,
equal_wf,
grp_car_wf,
eqtt_to_assert,
assert_of_mon_eq,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
istype-assert,
istype-void,
set_car_wf,
abdmonoid_wf,
loset_wf,
nil_in_oalist,
cons_wf,
nil_wf,
map_cons_lemma,
map_nil_lemma,
sd_ordered_cons_lemma,
before_nil_lemma,
sd_ordered_nil_lemma,
mem_cons_lemma,
mem_nil_lemma,
sd_ordered_wf,
map_wf,
pi1_wf_top,
mem_wf,
dset_of_mon_wf,
subtype_rel_self,
dset_of_mon_wf0,
pi2_wf,
or_false_r,
bor_wf,
bfalse_wf,
false_wf,
assert_of_bor,
or_functionality_wrt_uiff2
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation_alt,
cut,
applyEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
independent_functionElimination,
productElimination,
independent_isectElimination,
independent_pairFormation,
equalityIstype,
functionIsType,
voidElimination,
dependent_functionElimination,
universeIsType,
dependent_set_memberEquality_alt,
productEquality,
independent_pairEquality,
isect_memberEquality_alt,
natural_numberEquality,
productIsType,
lambdaEquality_alt,
unionIsType,
unionEquality
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}k:|a|. \mforall{}v:|b|. (inj(k,v) \mmember{} |oal(a;b)|)
Date html generated:
2019_10_16-PM-01_07_36
Last ObjectModification:
2019_06_24-PM-00_12_07
Theory : polynom_2
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