Nuprl Lemma : rng_before_imp_before_all
∀g:OCMon. ∀r:CDRng. ∀k:|g|. ∀ps:|omral(g;r)|.
((↑before(k;map(λz.(fst(z));ps))) ⇒ (↑(∀bx(:|g|) ∈ map(λz.(fst(z));ps). (x <b k))))
Proof
Definitions occuring in Statement :
omralist: omral(g;r),
before: before(u;ps),
ball: ball,
map: map(f;as),
assert: ↑b,
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
cdrng: CDRng,
grp_blt: a <b b,
oset_of_ocmon: g↓oset,
ocmon: OCMon,
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,
ocmon: OCMon,
omon: OMon,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
abmonoid: AbMon,
mon: Mon,
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,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
omralist: omral(g;r),
grp_blt: a <b b
Lemmas referenced :
before_imp_before_all,
oset_of_ocmon_wf,
subtype_rel_sets,
abmonoid_wf,
ulinorder_wf,
grp_car_wf,
assert_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,
cdrng_wf,
ocmon_wf,
cdrng_is_abdmonoid
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
instantiate,
hypothesis,
because_Cache,
lambdaEquality,
productEquality,
setElimination,
rename,
cumulativity,
universeEquality,
functionEquality,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setEquality,
independent_pairFormation
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}k:|g|. \mforall{}ps:|omral(g;r)|.
((\muparrow{}before(k;map(\mlambda{}z.(fst(z));ps))) {}\mRightarrow{} (\muparrow{}(\mforall{}\msubb{}x(:|g|) \mmember{} map(\mlambda{}z.(fst(z));ps). (x <\msubb{} k))))
Date html generated:
2017_10_01-AM-10_04_58
Last ObjectModification:
2017_03_03-PM-01_09_49
Theory : polynom_3
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