Nuprl Lemma : lookup_merge
∀a:LOSet. ∀b:AbDMon. ∀k:|a|. ∀ps,qs:|oal(a;b)|. (((ps ++ qs)[k]) = ((ps[k]) * (qs[k])) ∈ |b|)
Proof
Definitions occuring in Statement :
oal_merge: ps ++ qs,
lookup: as[k],
oalist: oal(a;b),
infix_ap: x f y,
all: ∀x:A. B[x],
equal: s = t ∈ T,
abdmonoid: AbDMon,
grp_id: e,
grp_op: *,
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],
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
so_lambda: λ2x y.t[x; y],
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
infix_ap: x f y,
so_apply: x[s1;s2],
implies: P ⇒ Q,
guard: {T},
so_lambda: λ2x.t[x],
prop: ℙ,
subtype_rel: A ⊆r B,
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,
so_apply: x[s],
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
oal_cons_pr: oal_cons_pr(x;y;ws),
rev_uimplies: rev_uimplies(P;Q),
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
cand: A c∧ B,
grp_car: |g|,
band_mon: <𝔹,∧b>,
grp_op: *,
pi2: snd(t)
Lemmas referenced :
set_car_wf,
abdmonoid_wf,
loset_wf,
oalist_pr_length_ind,
equal_wf,
grp_car_wf,
lookup_wf,
grp_id_wf,
oal_merge_wf,
grp_op_wf,
list_wf,
oalist_cases,
oalist_wf,
less_than_wf,
length_wf,
set_prod_wf,
dset_of_mon_wf,
nil_wf,
istype-less_than,
cons_wf,
istype-void,
istype-assert,
before_wf,
map_wf,
pi1_wf_top,
iff_weakening_equal,
subtype_rel_self,
imon_wf,
iabmonoid_wf,
abmonoid_wf,
subtype_rel_transitivity,
abdmonoid_abmonoid,
abmonoid_subtype_iabmonoid,
iabmonoid_subtype_imon,
mon_ident,
istype-universe,
true_wf,
squash_wf,
lookup_nil_lemma,
oal_merge_left_nil_lemma,
set_eq_wf,
ifthenelse_wf,
lookup_cons_pr_lemma,
oal_merge_right_nil_lemma,
oal_merge_conses_lemma,
set_blt_wf,
uiff_transitivity,
equal-wf-T-base,
bool_wf,
assert_wf,
set_lt_wf,
eqtt_to_assert,
assert_of_set_lt,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
set_lt_complement,
grp_eq_wf,
assert_of_mon_eq,
assert_of_dset_eq,
lookup_before_start,
oal_cons_pr_wf,
map_cons_lemma,
before_cons_lemma,
poset_sig_wf,
length_wf_nat,
length_of_cons_lemma,
nat_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
set_leq_antisymmetry,
set_leq_transitivity,
set_leq_weakening_eq,
lookup_before_start_a,
oal_merge_dom_pred,
sd_ordered_cons_lemma,
assert_of_band,
sd_ordered_wf,
assert_functionality_wrt_uiff,
mon_htfor_wf,
band_mon_wf,
ball_wf,
mon_subtype_grp_sig,
abmonoid_subtype_mon,
mon_wf,
grp_sig_wf,
sd_ordered_char,
mon_htfor_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
sqequalRule,
lambdaEquality_alt,
because_Cache,
applyEquality,
inhabitedIsType,
productEquality,
independent_functionElimination,
functionEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
voidEquality,
functionIsType,
equalityIstype,
independent_pairEquality,
productElimination,
Error :memTop,
baseClosed,
imageMemberEquality,
natural_numberEquality,
independent_isectElimination,
instantiate,
universeEquality,
imageElimination,
voidElimination,
isect_memberEquality_alt,
unionElimination,
equalityElimination,
independent_pairFormation,
applyLambdaEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}k:|a|. \mforall{}ps,qs:|oal(a;b)|. (((ps ++ qs)[k]) = ((ps[k]) * (qs[k])))
Date html generated:
2020_05_20-AM-09_36_03
Last ObjectModification:
2020_01_08-PM-06_26_10
Theory : polynom_2
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