Nuprl Lemma : omral_times_sd_ordered
∀g:OCMon. ∀r:CDRng. ∀ps,qs:(|g| × |r|) List.
((↑sd_ordered(map(λz.(fst(z));qs))) ⇒ (↑sd_ordered(map(λz.(fst(z));ps ** qs))))
Proof
Definitions occuring in Statement :
omral_times: ps ** qs,
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],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
pi1: fst(t),
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
so_apply: x[s],
omral_times: ps ** qs,
ycomb: Y,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
pi2: snd(t),
guard: {T}
Lemmas referenced :
list_induction,
grp_car_wf,
rng_car_wf,
all_wf,
list_wf,
assert_wf,
sd_ordered_wf,
oset_of_ocmon_wf,
ocmon_subtype_omon,
map_wf,
set_car_wf,
oset_of_ocmon_wf0,
omral_times_wf,
list_ind_nil_lemma,
istype-void,
map_nil_lemma,
sd_ordered_nil_lemma,
cdrng_wf,
ocmon_wf,
list_ind_cons_lemma,
omral_plus_sd_ordered,
omral_scale_wf,
omral_scale_sd_ordered
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
productEquality,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality_alt,
functionEquality,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
productElimination,
productIsType,
universeIsType,
inhabitedIsType,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
functionIsType
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}ps,qs:(|g| \mtimes{} |r|) List.
((\muparrow{}sd\_ordered(map(\mlambda{}z.(fst(z));qs))) {}\mRightarrow{} (\muparrow{}sd\_ordered(map(\mlambda{}z.(fst(z));ps ** qs))))
Date html generated:
2019_10_16-PM-01_09_02
Last ObjectModification:
2018_10_08-PM-00_42_08
Theory : polynom_3
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