Nuprl Lemma : r2-left-cases
∀a,b:ℝ^2. ∀c:{c:ℝ^2| |a + r(-1)*b⋅c + r(-1)*b| < (||a + r(-1)*b|| * ||c + r(-1)*b||)} .
  (r2-left(a;b;c) ∨ r2-left(a;c;b))
Proof
Definitions occuring in Statement : 
r2-left: r2-left(p;q;r)
, 
real-vec-norm: ||x||
, 
dot-product: x⋅y
, 
real-vec-mul: a*X
, 
real-vec-add: X + Y
, 
real-vec: ℝ^n
, 
rless: x < y
, 
rabs: |x|
, 
rmul: a * b
, 
int-to-real: r(n)
, 
all: ∀x:A. B[x]
, 
or: P ∨ Q
, 
set: {x:A| B[x]} 
, 
minus: -n
, 
natural_number: $n
Definitions unfolded in proof : 
rneq: x ≠ y
, 
eq_int: (i =z j)
, 
sq_exists: ∃x:A [B[x]]
, 
rless: x < y
, 
nequal: a ≠ b ∈ T 
, 
sq_stable: SqStable(P)
, 
req_int_terms: t1 ≡ t2
, 
real-vec-add: X + Y
, 
real-vec-mul: a*X
, 
real-vec-sub: X - Y
, 
req-vec: req-vec(n;x;y)
, 
assert: ↑b
, 
bnot: ¬bb
, 
sq_type: SQType(T)
, 
bfalse: ff
, 
lelt: i ≤ j < k
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
it: ⋅
, 
unit: Unit
, 
bool: 𝔹
, 
int_seg: {i..j-}
, 
real-vec: ℝ^n
, 
rminus: -(x)
, 
guard: {T}
, 
top: Top
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
int-to-real: r(n)
, 
decidable: Dec(P)
, 
true: True
, 
squash: ↓T
, 
less_than: a < b
, 
nat_plus: ℕ+
, 
real: ℝ
, 
subtype_rel: A ⊆r B
, 
rev_implies: P 
⇐ Q
, 
iff: P 
⇐⇒ Q
, 
uimplies: b supposing a
, 
or: P ∨ Q
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
implies: P 
⇒ Q
, 
not: ¬A
, 
false: False
, 
less_than': less_than'(a;b)
, 
and: P ∧ Q
, 
le: A ≤ B
, 
nat: ℕ
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
member: t ∈ T
, 
r2-left: r2-left(p;q;r)
, 
all: ∀x:A. B[x]
Lemmas referenced : 
rabs-positive-iff, 
real_term_value_minus_lemma, 
rless-implies-rless, 
nat_plus_properties, 
rnexp2, 
real-vec-norm-squared, 
rnexp-rmul, 
r2-dot-product, 
rnexp_functionality, 
req_transitivity, 
rnexp-rless, 
rabs-of-nonneg, 
rabs-rnexp, 
req_inversion, 
rnexp0, 
rnexp2-nonneg, 
rnexp_wf, 
zero-rleq-rabs, 
square-rless-implies, 
real-vec-norm_functionality, 
rmul_functionality, 
dot-product_functionality, 
sq_stable__rless, 
real_term_value_const_lemma, 
real_term_value_mul_lemma, 
real_term_value_var_lemma, 
real_term_value_add_lemma, 
real_term_value_sub_lemma, 
real_polynomial_null, 
req-iff-rsub-is-0, 
itermSubtract_wf, 
rsub_wf, 
radd_wf, 
r2-det-is-dot-product, 
rabs_functionality, 
neg_assert_of_eq_int, 
assert-bnot, 
bool_subtype_base, 
subtype_base_sq, 
bool_cases_sqequal, 
eqff_to_assert, 
lelt_wf, 
int_seg_wf, 
subtype_rel_self, 
assert_of_eq_int, 
eqtt_to_assert, 
bool_wf, 
eq_int_wf, 
real-vec-sub_wf, 
int_term_value_minus_lemma, 
itermMinus_wf, 
int_formula_prop_wf, 
int_formula_prop_eq_lemma, 
int_term_value_var_lemma, 
int_term_value_constant_lemma, 
int_term_value_mul_lemma, 
int_term_value_add_lemma, 
int_formula_prop_less_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_and_lemma, 
intformeq_wf, 
itermVar_wf, 
itermConstant_wf, 
itermMultiply_wf, 
itermAdd_wf, 
intformless_wf, 
intformnot_wf, 
intformand_wf, 
full-omega-unsat, 
rless-iff2, 
decidable__lt, 
int-value-type, 
set-value-type, 
equal_wf, 
less_than_wf, 
real_wf, 
req_weakening, 
rless_functionality, 
or_wf, 
rminus_wf, 
r2-det_wf, 
r2-det-antisymmetry, 
real-vec-norm_wf, 
rmul_wf, 
int-to-real_wf, 
real-vec-mul_wf, 
real-vec-add_wf, 
dot-product_wf, 
rabs_wf, 
rless_wf, 
le_wf, 
false_wf, 
real-vec_wf, 
set_wf
Rules used in proof : 
imageElimination, 
cumulativity, 
instantiate, 
promote_hyp, 
functionEquality, 
equalityElimination, 
inrFormation, 
voidEquality, 
voidElimination, 
isect_memberEquality, 
int_eqEquality, 
approximateComputation, 
multiplyEquality, 
addEquality, 
dependent_pairFormation, 
inlFormation, 
unionElimination, 
equalitySymmetry, 
equalityTransitivity, 
cutEval, 
baseClosed, 
imageMemberEquality, 
applyEquality, 
intEquality, 
independent_functionElimination, 
productElimination, 
independent_isectElimination, 
dependent_functionElimination, 
orFunctionality, 
addLevel, 
rename, 
setElimination, 
minusEquality, 
because_Cache, 
lambdaEquality, 
hypothesisEquality, 
hypothesis, 
independent_pairFormation, 
sqequalRule, 
natural_numberEquality, 
dependent_set_memberEquality, 
thin, 
isectElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
cut, 
lambdaFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution
Latex:
\mforall{}a,b:\mBbbR{}\^{}2.  \mforall{}c:\{c:\mBbbR{}\^{}2|  |a  +  r(-1)*b\mcdot{}c  +  r(-1)*b|  <  (||a  +  r(-1)*b||  *  ||c  +  r(-1)*b||)\}  .
    (r2-left(a;b;c)  \mvee{}  r2-left(a;c;b))
Date html generated:
2018_05_22-PM-02_38_24
Last ObjectModification:
2018_05_21-AM-00_50_59
Theory : reals
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