Nuprl Lemma : lookup_omral_scale_c
∀g:OCMon. ∀r:CDRng. ∀z,k:|g|. ∀v:|r|. ∀ps:|omral(g;r)|.
(((<k,v>* ps)[z]) = (msFor{r↓+gp} y ∈ dom(ps). when (k * y) =b z. (v * (ps[y]))) ∈ |r|)
Proof
Definitions occuring in Statement :
omral_scale: <k,v>* ps,
omral_dom: dom(ps),
omralist: omral(g;r),
lookup: as[k],
mset_for: mset_for,
infix_ap: x f y,
all: ∀x:A. B[x],
equal: s = t ∈ T,
rng_when: rng_when,
add_grp_of_rng: r↓+gp,
cdrng: CDRng,
rng_times: *,
rng_zero: 0,
rng_car: |r|,
oset_of_ocmon: g↓oset,
ocmon: OCMon,
grp_op: *,
grp_eq: =b,
grp_car: |g|,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
dset: DSet,
cdrng: CDRng,
crng: CRng,
rng: Rng,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
decidable: Dec(P),
or: P ∨ Q,
omralist: omral(g;r),
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,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
add_grp_of_rng: r↓+gp,
grp_car: |g|,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
omon: OMon,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
bfalse: ff,
infix_ap: x f y,
so_apply: x[s],
cand: A c∧ B,
omral_dom: dom(ps),
null_mset: 0{s},
oal_dom: dom(ps),
mk_mset: mk_mset(as),
top: Top,
assert: ↑b,
not: ¬A,
exists: ∃x:A. B[x],
false: False,
mset_mem: mset_mem,
mem: a ∈b as,
set_eq: =b,
pi2: snd(t),
mon_for: For{g} x ∈ as. f[x],
bor_mon: <𝔹,∨b>,
grp_op: *,
grp_id: e,
for: For{T,op,id} x ∈ as. f[x],
tlambda: λx:T. b[x],
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rng_when: rng_when,
squash: ↓T,
true: True,
abgrp: AbGrp,
grp: Group{i},
iabmonoid: IAbMonoid,
imon: IMonoid,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
set_car_wf,
omralist_wf,
dset_wf,
rng_car_wf,
grp_car_wf,
cdrng_wf,
ocmon_wf,
list_induction,
decidable_wf,
exists_wf,
assert_wf,
mset_mem_wf,
oset_of_ocmon_wf,
subtype_rel_sets,
abmonoid_wf,
ulinorder_wf,
infix_ap_wf,
bool_wf,
grp_le_wf,
equal_wf,
grp_eq_wf,
eqtt_to_assert,
cancel_wf,
grp_op_wf,
uall_wf,
monot_wf,
omral_dom_wf,
list_wf,
map_nil_lemma,
mset_mem_null_lemma,
false_wf,
map_cons_lemma,
for_cons_lemma,
reduce_wf,
bor_wf,
bfalse_wf,
map_wf,
pi1_wf,
assert_of_bor,
or_wf,
cons_wf,
not_wf,
decidable__mon_eq,
abdmonoid_dmon,
ocmon_subtype_abdmonoid,
subtype_rel_transitivity,
abdmonoid_wf,
dmon_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_mon_eq,
and_wf,
squash_wf,
true_wf,
lookup_wf,
poset_sig_wf,
oset_of_ocmon_wf0,
rng_zero_wf,
omral_scale_wf,
lookup_omral_scale_a,
mset_for_functionality,
add_grp_of_rng_wf_b,
grp_sig_wf,
monoid_p_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
comm_wf,
set_wf,
mon_when_wf,
add_grp_of_rng_wf_a,
dset_of_mon_wf0,
add_grp_of_rng_wf,
rng_times_wf,
imon_wf,
fset_for_when_unique,
omral_dom_wf2,
iff_weakening_equal,
ocmon_cancel,
lookup_omral_scale_b,
mset_for_when_none
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
unionElimination,
productEquality,
because_Cache,
instantiate,
cumulativity,
universeEquality,
functionEquality,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setEquality,
independent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
inrFormation,
functionExtensionality,
inlFormation,
dependent_pairFormation,
independent_pairEquality,
addLevel,
impliesFunctionality,
orFunctionality,
dependent_set_memberEquality,
applyLambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}z,k:|g|. \mforall{}v:|r|. \mforall{}ps:|omral(g;r)|.
(((<k,v>* ps)[z]) = (msFor\{r\mdownarrow{}+gp\} y \mmember{} dom(ps). when (k * y) =\msubb{} z. (v * (ps[y]))))
Date html generated:
2017_10_01-AM-10_06_02
Last ObjectModification:
2017_03_03-PM-01_13_27
Theory : polynom_3
Home
Index