Nuprl Lemma : r2-left-right
∀a,b,x,y:ℝ^2. (r2-left(x;a;b) ⇒ r2-left(y;b;a) ⇒ (∃z:ℝ^2. (rv-T(2;x;z;y) ∧ (¬rv-pos-angle(2;z;a;b)))))
Proof
Definitions occuring in Statement :
r2-left: r2-left(p;q;r),
rv-T: rv-T(n;a;b;c),
rv-pos-angle: rv-pos-angle(n;a;b;c),
real-vec: ℝ^n,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
cand: A c∧ B,
rv-T: rv-T(n;a;b;c),
real-vec-be: real-vec-be(n;a;b;c),
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
real-vec-mul: a*X,
real-vec-add: X + Y,
req_int_terms: t1 ≡ t2,
top: Top,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rv-pos-angle: rv-pos-angle(n;a;b;c),
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
eq_int: (i =z j),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
rneq: x ≠ y,
rdiv: (x/y)
Lemmas referenced :
r2-left-right-lemma,
real-vec-add_wf,
false_wf,
le_wf,
real-vec-mul_wf,
rsub_wf,
int-to-real_wf,
rv-T_wf,
not_wf,
rv-pos-angle_wf,
r2-left_wf,
real-vec_wf,
real-vec-sep_wf,
i-member_wf,
rccint_wf,
req-vec_wf,
equal_wf,
req_weakening,
int_seg_wf,
not-real-vec-sep-iff-eq,
req-vec_functionality,
real-vec-add_functionality,
req-vec_weakening,
real-vec-mul_functionality,
radd_wf,
rmul_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
r2-det_wf,
dot-product_wf,
real-vec-sub_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
rminus_wf,
lelt_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
req_wf,
req_functionality,
r2-det-is-dot-product,
rless_wf,
rabs_wf,
real-vec-norm_wf,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
r2-dot-product,
Cauchy-Schwarz-equality2,
real-vec-norm-positive-iff,
decidable__equal_int,
int_subtype_base,
int_seg_properties,
int_seg_subtype,
int_seg_cases,
intformand_wf,
intformless_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
rdiv_wf,
rmul_preserves_req,
rinv_wf2,
req-implies-req,
itermMinus_wf,
req_transitivity,
rmul_functionality,
rmul-rinv,
rmul-rinv3,
real_term_value_minus_lemma,
radd-preserves-req,
radd-zero,
req_inversion
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
dependent_pairFormation,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
because_Cache,
productEquality,
voidElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
independent_isectElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
setElimination,
rename,
unionElimination,
equalityElimination,
imageMemberEquality,
baseClosed,
promote_hyp,
instantiate,
cumulativity,
functionEquality,
addLevel,
impliesFunctionality,
hypothesis_subsumption,
addEquality,
inrFormation,
inlFormation
Latex:
\mforall{}a,b,x,y:\mBbbR{}\^{}2.
(r2-left(x;a;b) {}\mRightarrow{} r2-left(y;b;a) {}\mRightarrow{} (\mexists{}z:\mBbbR{}\^{}2. (rv-T(2;x;z;y) \mwedge{} (\mneg{}rv-pos-angle(2;z;a;b)))))
Date html generated:
2017_10_03-AM-11_56_56
Last ObjectModification:
2017_06_09-PM-06_49_10
Theory : reals
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