Nuprl Lemma : closures-meet-sq-ext
∀[P,Q:ℝ ⟶ ℙ].
((∃a:{a:ℝ| P a} . (∃b:ℝ [((Q b) ∧ (a ≤ b))]))
⇒ (∃c:{c:ℝ| (r0 ≤ c) ∧ (c < r1)}
∀a:{a:ℝ| P a} . ∀b:{b:ℝ| (Q b) ∧ (a ≤ b)} .
∃a':{a':ℝ| P a'} . (∃b':{b':ℝ| (Q b') ∧ (a' ≤ b')} [((a ≤ a') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c)))]))
⇒ (∃y:ℝ. (y ∈ closure(λz.(↓P z)) ∧ y ∈ closure(λz.(↓Q z)))))
Proof
Definitions occuring in Statement :
member-closure: y ∈ closure(A),
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
pi1: fst(t),
so_apply: x[s],
so_lambda: λ2x.t[x],
pi2: snd(t),
subtract: n - m,
rabs: |x|,
rmax: rmax(x;y),
imax: imax(a;b),
ifthenelse: if b then t else f fi ,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
btrue: tt,
it: ⋅,
bfalse: ff,
canonical-bound: canonical-bound(r),
divide: n ÷ m,
absval: |i|,
int-to-real: r(n),
let: let,
closures-meet-sq,
common-limit-squeeze-ext,
sq_stable__rleq,
converges-to_functionality,
rmul-limit,
constant-limit,
req_weakening,
rpowers-converge-ext,
rless_functionality,
sq_stable__rless,
integer-bound,
converges-implies-bounded,
rleq_functionality_wrt_implies,
sq-all-large-and,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
top: Top,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
false: False
Lemmas referenced :
closures-meet-sq,
lifting-strict-spread,
istype-void,
strict4-apply,
strict4-spread,
value-type-has-value,
int-value-type,
has-value_wf_base,
istype-base,
is-exception_wf,
istype-universe,
strict4-divide,
lifting-strict-callbyvalue,
lifting-strict-decide,
strict4-decide,
cbv_sqequal,
lifting-strict-less,
exception-not-value,
common-limit-squeeze-ext,
sq_stable__rleq,
converges-to_functionality,
rmul-limit,
constant-limit,
req_weakening,
rpowers-converge-ext,
rless_functionality,
sq_stable__rless,
integer-bound,
converges-implies-bounded,
rleq_functionality_wrt_implies,
sq-all-large-and
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination,
independent_pairFormation,
lambdaFormation_alt,
callbyvalueAdd,
baseApply,
closedConclusion,
hypothesisEquality,
productElimination,
intEquality,
universeIsType,
addExceptionCases,
exceptionSqequal,
inrFormation_alt,
imageMemberEquality,
imageElimination,
inlFormation_alt,
callbyvalueCallbyvalue,
callbyvalueReduce,
callbyvalueExceptionCases,
because_Cache,
callbyvalueMultiply,
multiplyExceptionCases,
sqleReflexivity,
independent_functionElimination
Latex:
\mforall{}[P,Q:\mBbbR{} {}\mrightarrow{} \mBbbP{}].
((\mexists{}a:\{a:\mBbbR{}| P a\} . (\mexists{}b:\mBbbR{} [((Q b) \mwedge{} (a \mleq{} b))]))
{}\mRightarrow{} (\mexists{}c:\{c:\mBbbR{}| (r0 \mleq{} c) \mwedge{} (c < r1)\}
\mforall{}a:\{a:\mBbbR{}| P a\} . \mforall{}b:\{b:\mBbbR{}| (Q b) \mwedge{} (a \mleq{} b)\} .
\mexists{}a':\{a':\mBbbR{}| P a'\} . (\mexists{}b':\{b':\mBbbR{}| (Q b') \mwedge{} (a' \mleq{} b')\} [((a \mleq{} a') \mwedge{} (b' \mleq{} b) \mwedge{} ((b' - a') \mleq{} ((\000Cb - a) * c)))]))
{}\mRightarrow{} (\mexists{}y:\mBbbR{}. (y \mmember{} closure(\mlambda{}z.(\mdownarrow{}P z)) \mwedge{} y \mmember{} closure(\mlambda{}z.(\mdownarrow{}Q z)))))
Date html generated:
2019_10_29-AM-10_42_10
Last ObjectModification:
2019_04_05-PM-05_08_37
Theory : reals
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