Nuprl Lemma : omral_scale_dom_bound
∀g:OCMon. ∀r:CDRng. ∀bound,k:|g|. ∀v:|r|. ∀ps:(|g| × |r|) List.
((↑(∀bx(:|g|) ∈ map(λz.(fst(z));ps)
(x <b bound)))
⇒ (↑(∀bx(:|g|) ∈ map(λz.(fst(z));<k,v>* ps)
(x <b (k * bound)))))
Proof
Definitions occuring in Statement :
omral_scale: <k,v>* ps,
ball: ball,
map: map(f;as),
list: T List,
assert: ↑b,
infix_ap: x f y,
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|,
grp_blt: a <b b,
ocmon: OCMon,
grp_op: *,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
infix_ap: x f y,
so_apply: x[s],
prop: ℙ,
cdrng: CDRng,
crng: CRng,
rng: Rng,
pi1: fst(t),
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
subtype_rel: A ⊆r B,
omon: OMon,
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),
uimplies: b supposing a,
bfalse: ff,
cand: A c∧ B,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
omral_scale_dom_pred,
grp_blt_wf,
grp_op_wf,
grp_car_wf,
assert_wf,
ball_wf,
map_wf,
rng_car_wf,
list_wf,
cdrng_wf,
ocmon_wf,
iff_transitivity,
infix_ap_wf,
all_wf,
mem_wf,
oset_of_ocmon_wf,
subtype_rel_sets,
abmonoid_wf,
ulinorder_wf,
bool_wf,
grp_le_wf,
equal_wf,
grp_eq_wf,
eqtt_to_assert,
cancel_wf,
uall_wf,
monot_wf,
iff_weakening_uiff,
assert_functionality_wrt_uiff,
squash_wf,
true_wf,
ball_char,
set_car_wf,
oset_of_ocmon_wf0,
pi1_wf,
assert_of_grp_blt,
grp_op_preserves_lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
isectElimination,
setElimination,
rename,
hypothesis,
applyEquality,
because_Cache,
independent_functionElimination,
productEquality,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
instantiate,
cumulativity,
universeEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
setEquality,
independent_pairFormation,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairEquality
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}bound,k:|g|. \mforall{}v:|r|. \mforall{}ps:(|g| \mtimes{} |r|) List.
((\muparrow{}(\mforall{}\msubb{}x(:|g|) \mmember{} map(\mlambda{}z.(fst(z));ps)
(x <\msubb{} bound)))
{}\mRightarrow{} (\muparrow{}(\mforall{}\msubb{}x(:|g|) \mmember{} map(\mlambda{}z.(fst(z));<k,v>* ps)
(x <\msubb{} (k * bound)))))
Date html generated:
2017_10_01-AM-10_05_32
Last ObjectModification:
2017_03_03-PM-01_11_56
Theory : polynom_3
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