Nuprl Lemma : by-nearby-cases-ext
∀[P:ℝ ⟶ ℝ ⟶ ℙ]
∀n:ℕ+. ∀x:ℝ.
((∀y:{y:ℝ| x < y} . P[x;y])
⇒ (∀y:{y:ℝ| y < x} . P[x;y])
⇒ (∀y:{y:ℝ| |x - y| ≤ (r1/r(n))} . P[x;y])
⇒ (∀y:ℝ. P[x;y]))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
by-nearby-cases,
nearby-cases-ext,
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],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T
Lemmas referenced :
by-nearby-cases,
lifting-strict-callbyvalue,
top_wf,
equal_wf,
has-value_wf_base,
base_wf,
is-exception_wf,
lifting-strict-less,
nearby-cases-ext
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
isectElimination,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueDecide,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
unionEquality,
unionElimination,
sqleReflexivity,
dependent_functionElimination,
independent_functionElimination,
baseApply,
closedConclusion,
decideExceptionCases,
inrFormation,
because_Cache,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation
Latex:
\mforall{}[P:\mBbbR{} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbP{}]
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}x:\mBbbR{}.
((\mforall{}y:\{y:\mBbbR{}| x < y\} . P[x;y])
{}\mRightarrow{} (\mforall{}y:\{y:\mBbbR{}| y < x\} . P[x;y])
{}\mRightarrow{} (\mforall{}y:\{y:\mBbbR{}| |x - y| \mleq{} (r1/r(n))\} . P[x;y])
{}\mRightarrow{} (\mforall{}y:\mBbbR{}. P[x;y]))
Date html generated:
2017_10_03-AM-08_48_26
Last ObjectModification:
2017_07_28-AM-07_33_25
Theory : reals
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