Nuprl Lemma : omral_scale_dom_pred
∀g:OCMon. ∀r:CDRng. ∀Q:|g| ⟶ 𝔹. ∀k:|g|. ∀v:|r|. ∀ps:(|g| × |r|) List.
((↑(∀bx(:|g|) ∈ map(λz.(fst(z));ps). Q[k * x])) ⇒ (↑(∀bx(:|g|) ∈ map(λz.(fst(z));<k,v>* ps). Q[x])))
Proof
Definitions occuring in Statement :
omral_scale: <k,v>* ps,
ball: ball,
map: map(f;as),
list: T List,
assert: ↑b,
bool: 𝔹,
infix_ap: x f y,
so_apply: x[s],
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
cdrng: CDRng,
rng_car: |r|,
ocmon: OCMon,
grp_op: *,
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,
pi1: fst(t),
so_apply: x[s],
infix_ap: x f y,
ball: ball,
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,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
or: P ∨ Q,
band: p ∧b q,
bfalse: ff,
false: False,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
pi2: snd(t),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
list_induction,
grp_car_wf,
rng_car_wf,
assert_wf,
ball_wf,
map_wf,
infix_ap_wf,
grp_op_wf,
omral_scale_wf,
list_wf,
map_nil_lemma,
list_ind_nil_lemma,
ball_nil_lemma,
true_wf,
map_cons_lemma,
ball_cons_lemma,
bool_cases_sqequal,
pi1_wf,
bool_wf,
eqtt_to_assert,
equal_wf,
cdrng_wf,
ocmon_wf,
list_ind_cons_lemma,
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,
assert_of_band,
and_wf,
assert_elim,
subtype_base_sq,
bool_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
productEquality,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
functionEquality,
dependent_functionElimination,
productElimination,
hypothesisEquality,
applyEquality,
functionExtensionality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
independent_pairEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
independent_pairFormation,
impliesFunctionality,
dependent_set_memberEquality,
applyLambdaEquality,
instantiate,
cumulativity
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}Q:|g| {}\mrightarrow{} \mBbbB{}. \mforall{}k:|g|. \mforall{}v:|r|. \mforall{}ps:(|g| \mtimes{} |r|) List.
((\muparrow{}(\mforall{}\msubb{}x(:|g|) \mmember{} map(\mlambda{}z.(fst(z));ps)
Q[k * x]))
{}\mRightarrow{} (\muparrow{}(\mforall{}\msubb{}x(:|g|) \mmember{} map(\mlambda{}z.(fst(z));<k,v>* ps)
Q[x])))
Date html generated:
2017_10_01-AM-10_05_29
Last ObjectModification:
2017_03_03-PM-01_11_24
Theory : polynom_3
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