Nuprl Lemma : omral_scale_sd_ordered
∀g:OCMon. ∀r:CDRng. ∀k:|g|. ∀v:|r|. ∀ps:(|g| × |r|) List.
((↑sd_ordered(map(λz.(fst(z));ps))) ⇒ (↑sd_ordered(map(λz.(fst(z));<k,v>* ps))))
Proof
Definitions occuring in Statement :
omral_scale: <k,v>* ps,
sd_ordered: sd_ordered(as),
map: map(f;as),
list: T List,
assert: ↑b,
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
product: x:A × B[x],
cdrng: CDRng,
rng_car: |r|,
oset_of_ocmon: g↓oset,
ocmon: OCMon,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
cdrng: CDRng,
crng: CRng,
rng: Rng,
so_lambda: λ2x.t[x],
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
omon: OMon,
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),
uimplies: b supposing a,
bfalse: ff,
infix_ap: x f y,
so_apply: x[s],
cand: A c∧ B,
pi1: fst(t),
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
top: Top,
omral_scale: <k,v>* ps,
ycomb: Y,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
assert: ↑b,
true: True,
pi2: snd(t),
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
band_mon: <𝔹,∧b>,
grp_car: |g|,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
grp_op: *,
ball: ball,
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),
nil: [],
grp_id: e,
grp_lt: a < b
Lemmas referenced :
cdrng_is_abdmonoid,
list_induction,
grp_car_wf,
rng_car_wf,
assert_wf,
sd_ordered_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,
map_wf,
set_car_wf,
oset_of_ocmon_wf0,
omral_scale_wf,
list_wf,
map_nil_lemma,
list_ind_nil_lemma,
sd_ordered_nil_lemma,
map_cons_lemma,
sd_ordered_cons_lemma,
cdrng_wf,
ocmon_wf,
true_wf,
list_ind_cons_lemma,
assert_of_band,
before_wf,
rng_eq_wf,
rng_times_wf,
rng_zero_wf,
uiff_transitivity,
equal-wf-T-base,
assert_of_rng_eq,
cdrng_subtype_drng,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
cons_wf,
mon_htfor_wf,
band_mon_wf,
ball_wf,
set_blt_wf,
pi1_wf,
assert_functionality_wrt_uiff,
sd_ordered_char,
mon_htfor_cons_lemma,
nil_wf,
ball_cons_lemma,
assert_of_set_lt,
set_lt_wf,
grp_op_preserves_lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productEquality,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
productElimination,
lambdaEquality,
functionEquality,
dependent_functionElimination,
applyEquality,
instantiate,
cumulativity,
universeEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setEquality,
independent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
baseClosed,
impliesFunctionality,
independent_pairEquality
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}k:|g|. \mforall{}v:|r|. \mforall{}ps:(|g| \mtimes{} |r|) List.
((\muparrow{}sd\_ordered(map(\mlambda{}z.(fst(z));ps))) {}\mRightarrow{} (\muparrow{}sd\_ordered(map(\mlambda{}z.(fst(z));<k,v>* ps))))
Date html generated:
2017_10_01-AM-10_05_38
Last ObjectModification:
2017_03_03-PM-01_12_56
Theory : polynom_3
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