Nuprl Lemma : by-nearby-cases
∀[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 :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s]
Lemmas referenced :
nearby-cases-ext,
real_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
isectElimination,
setEquality,
because_Cache,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
sqequalRule,
inrFormation,
productElimination,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
applyEquality,
functionExtensionality,
functionEquality,
cumulativity,
universeEquality,
dependent_set_memberEquality
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:
2016_10_26-AM-09_12_14
Last ObjectModification:
2016_10_13-AM-10_58_06
Theory : reals
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