Nuprl Lemma : omral_bilinear
∀g:OCMon. ∀a:CDRng. BiLinear(|omral(g;a)|;λps,qs. (ps ++ qs);λps,qs. (ps ** qs))
Proof
Definitions occuring in Statement :
omral_times: ps ** qs,
omral_plus: ps ++ qs,
omralist: omral(g;r),
all: ∀x:A. B[x],
lambda: λx.A[x],
cdrng: CDRng,
ocmon: OCMon,
set_car: |p|,
bilinear: BiLinear(T;pl;tm)
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
dset: DSet,
uimplies: b supposing a,
infix_ap: x f y,
implies: P ⇒ Q,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
squash: ↓T,
prop: ℙ,
cdrng: CDRng,
crng: CRng,
rng: Rng,
rng_mssum: rng_mssum,
omon: OMon,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
so_apply: x[s],
cand: A c∧ B,
abgrp: AbGrp,
grp: Group{i},
iabmonoid: IAbMonoid,
imon: IMonoid,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
add_grp_of_rng: r↓+gp,
grp_car: |g|,
omralist: omral(g;r),
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
dset_list: s List,
set_prod: s × t,
grp_id: e,
pi2: snd(t),
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
label: ...$L... t,
rng_car: |r|,
mset_for: mset_for,
mon_for: For{g} x ∈ as. f[x],
for: For{T,op,id} x ∈ as. f[x],
reduce: reduce(f;k;as),
list_ind: list_ind,
map: map(f;as),
omral_dom: dom(ps),
oal_dom: dom(ps),
mk_mset: mk_mset(as),
omral_plus: ps ++ qs,
oal_merge: ps ++ qs,
ycomb: Y,
null: null(as),
rev_uimplies: rev_uimplies(P;Q),
or: P ∨ Q
Lemmas referenced :
bilinear_comm_elim,
set_car_wf,
omralist_wf,
dset_wf,
omral_plus_wf2,
omral_times_wf2,
cdrng_wf,
ocmon_wf,
omral_times_comm,
omral_lookups_same_a,
grp_car_wf,
equal_wf,
squash_wf,
true_wf,
rng_car_wf,
lookup_omral_times,
rng_mssum_functionality_wrt_equal,
oset_of_ocmon_wf,
subtype_rel_sets,
abmonoid_wf,
ulinorder_wf,
assert_wf,
infix_ap_wf,
bool_wf,
grp_le_wf,
grp_eq_wf,
eqtt_to_assert,
cancel_wf,
grp_op_wf,
uall_wf,
monot_wf,
mset_for_wf,
add_grp_of_rng_wf_b,
grp_sig_wf,
monoid_p_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
comm_wf,
set_wf,
rng_when_wf,
oset_of_ocmon_wf0,
dset_of_mon_wf0,
add_grp_of_rng_wf,
rng_times_wf,
lookup_wf,
rng_zero_wf,
omral_plus_wf,
list_wf,
omral_dom_wf,
rng_mssum_wf,
rng_plus_wf,
rng_wf,
lookup_omral_plus,
mset_mem_wf,
iff_weakening_equal,
mset_union_wf,
mset_for_dom_shift,
omral_plus_dom,
mset_diff_wf,
lookup_omral_eq_zero,
assert_functionality_wrt_uiff,
bnot_wf,
mset_mem_diff,
mset_union_wf_f,
omral_dom_wf2,
iff_transitivity,
not_wf,
iff_weakening_uiff,
assert_of_band,
assert_of_bnot,
rng_times_zero,
rng_when_of_zero,
mem_bsubmset,
bor_wf,
fset_mem_union,
assert_of_bor,
rng_times_over_plus,
rng_when_thru_plus,
rng_mssum_of_plus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
because_Cache,
independent_isectElimination,
independent_functionElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
instantiate,
productEquality,
cumulativity,
functionEquality,
unionElimination,
equalityElimination,
productElimination,
setEquality,
independent_pairFormation,
natural_numberEquality,
imageMemberEquality,
baseClosed,
inlFormation,
inrFormation
Latex:
\mforall{}g:OCMon. \mforall{}a:CDRng. BiLinear(|omral(g;a)|;\mlambda{}ps,qs. (ps ++ qs);\mlambda{}ps,qs. (ps ** qs))
Date html generated:
2017_10_01-AM-10_06_35
Last ObjectModification:
2017_03_03-PM-01_16_15
Theory : polynom_3
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