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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