Nuprl Lemma : rv-circle-circle-lemma2
∀n:{2...}. ∀r1,r2:{r:ℝ| r0 ≤ r} . ∀b:ℝ^n.
((r0 < ||b||)
⇒ ((r1^2 - r2^2) + ||b||^2^2 ≤ (r(4) * ||b||^2 * r1^2))
⇒ (∃u,v:ℝ^n
(((||u|| = r1) ∧ (||u - b|| = r2))
∧ ((||v|| = r1) ∧ (||v - b|| = r2))
∧ (((r1^2 - r2^2) + ||b||^2^2 < (r(4) * ||b||^2 * r1^2)) ⇒ u ≠ v))))
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
real-vec-norm: ||x||,
real-vec-sub: X - Y,
real-vec: ℝ^n,
rleq: x ≤ y,
rless: x < y,
rnexp: x^k1,
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
real-vec-dist: d(x;y),
real-vec-sep: a ≠ b,
req-vec: req-vec(n;x;y),
real-vec-add: X + Y,
real-vec-sub: X - Y,
real-vec-mul: a*X,
let: let,
subtract: n - m,
primrec: primrec(n;b;c),
exp: i^n,
true: True,
less_than: a < b,
nat: ℕ,
rdiv: (x/y),
rneq: x ≠ y,
rev_implies: P ⇐ Q,
req_int_terms: t1 ≡ t2,
nequal: a ≠ b ∈ T ,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
sq_stable: SqStable(P),
real: ℝ,
so_apply: x[s],
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
so_lambda: λ2x.t[x],
dot-product: x⋅y,
cand: A c∧ B,
real-vec: ℝ^n,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
nat_plus: ℕ+,
sq_exists: ∃x:A [B[x]],
rless: x < y,
int_upper: {i...},
lelt: i ≤ j < k,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
decidable: Dec(P),
int_seg: {i..j-},
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
prop: ℙ,
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
radd-preserves-rless,
rnexp2,
req_witness,
sq_stable__req,
sq_stable__and,
square-rless-implies,
rabs-rmul,
sq_stable__rleq,
rleq_weakening_rless,
rleq-int,
rmul-is-positive,
real-vec-norm_functionality,
rmul_preserves_rless,
exists_wf,
real-vec-sep_wf,
req-vec_weakening,
or_wf,
all_wf,
real-vec-sub_wf,
req-vec_wf,
real-vec-add_wf,
set_wf,
rsqrt_wf,
rnexp-positive,
rmul-rinv3,
rdiv_functionality,
rnexp-rdiv,
req_inversion,
rless_functionality,
rnexp-int,
iff_weakening_equal,
subtype_rel_self,
nat_wf,
true_wf,
squash_wf,
rmul_comm,
rleq_weakening,
rless_transitivity1,
req-int,
exp_wf2,
radd-zero,
rless-int,
radd-preserves-rleq,
rnexp_wf,
rmul-rinv,
rleq_functionality,
real-vec-norm-nonneg,
rmul-one,
rinv_wf2,
rmul-zero-both,
rmul_preserves_rleq,
rabs-of-nonneg,
real-vec-norm-mul,
rabs_wf,
rmul_functionality,
dot-product-linearity2,
req_transitivity,
rdiv_wf,
real-vec-mul_wf,
rsub_wf,
rless-implies-rless,
rneq_wf,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
itermMinus_wf,
itermMultiply_wf,
rsum-zero,
le_wf,
rmul-zero,
rsum_functionality,
rsum-split-last,
radd_functionality,
req_weakening,
req_functionality,
int_term_value_add_lemma,
itermAdd_wf,
radd_wf,
rsum_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
decidable__le,
sq_stable__less_than,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
subtract-add-cancel,
rmul_wf,
subtract_wf,
rsum-split,
dot-product_wf,
req_wf,
int_seg_wf,
rminus_wf,
eq_int_wf,
ifthenelse_wf,
equal_wf,
equal-wf-base-T,
not_wf,
equal-wf-base,
int_formula_prop_eq_lemma,
intformeq_wf,
lelt_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
int_upper_properties,
nat_plus_properties,
int_seg_properties,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
real-vec-norm-positive-iff,
int_upper_wf,
rleq_wf,
real_wf,
real-vec_wf,
real-vec-norm_wf,
int-to-real_wf,
rless_wf,
false_wf,
upper_subtype_nat,
rv-circle-circle-lemma
Rules used in proof :
functionEquality,
universeEquality,
inlFormation,
inrFormation,
imageElimination,
imageMemberEquality,
addEquality,
promote_hyp,
equalityElimination,
productEquality,
baseClosed,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
lambdaEquality,
approximateComputation,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
dependent_pairFormation,
intEquality,
cumulativity,
instantiate,
unionElimination,
rename,
setElimination,
productElimination,
setEquality,
because_Cache,
independent_functionElimination,
independent_pairFormation,
sqequalRule,
independent_isectElimination,
natural_numberEquality,
isectElimination,
applyEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}n:\{2...\}. \mforall{}r1,r2:\{r:\mBbbR{}| r0 \mleq{} r\} . \mforall{}b:\mBbbR{}\^{}n.
((r0 < ||b||)
{}\mRightarrow{} ((r1\^{}2 - r2\^{}2) + ||b||\^{}2\^{}2 \mleq{} (r(4) * ||b||\^{}2 * r1\^{}2))
{}\mRightarrow{} (\mexists{}u,v:\mBbbR{}\^{}n
(((||u|| = r1) \mwedge{} (||u - b|| = r2))
\mwedge{} ((||v|| = r1) \mwedge{} (||v - b|| = r2))
\mwedge{} (((r1\^{}2 - r2\^{}2) + ||b||\^{}2\^{}2 < (r(4) * ||b||\^{}2 * r1\^{}2)) {}\mRightarrow{} u \mneq{} v))))
Date html generated:
2018_05_22-PM-02_38_02
Last ObjectModification:
2018_05_21-AM-00_58_09
Theory : reals
Home
Index