Nuprl Lemma : geo-general-position-implies
∀g:OrientedPlane. ∀xs:{xs:Point List| geo-general-position(g;xs)} . ∀i,j,k:ℕ||xs||.
  ((¬(i = j ∈ ℤ)) 
⇒ (¬(k = i ∈ ℤ)) 
⇒ (¬(k = j ∈ ℤ)) 
⇒ xs[i] # xs[j]xs[k])
Proof
Definitions occuring in Statement : 
geo-general-position: geo-general-position(g;xs)
, 
oriented-plane: OrientedPlane
, 
geo-lsep: a # bc
, 
geo-point: Point
, 
select: L[n]
, 
length: ||as||
, 
list: T List
, 
int_seg: {i..j-}
, 
all: ∀x:A. B[x]
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
set: {x:A| B[x]} 
, 
natural_number: $n
, 
int: ℤ
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
uimplies: b supposing a
, 
guard: {T}
, 
subtype_rel: A ⊆r B
, 
int_seg: {i..j-}
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
less_than: a < b
, 
top: Top
, 
false: False
, 
exists: ∃x:A. B[x]
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
not: ¬A
, 
or: P ∨ Q
, 
decidable: Dec(P)
, 
and: P ∧ Q
, 
lelt: i ≤ j < k
, 
oriented-plane: Error :oriented-plane, 
geo-general-position: geo-general-position(g;xs)
, 
ge: i ≥ j 
, 
nat: ℕ
, 
less_than': less_than'(a;b)
, 
le: A ≤ B
, 
cand: A c∧ B
, 
uiff: uiff(P;Q)
, 
assert: ↑b
, 
bnot: ¬bb
, 
sq_type: SQType(T)
, 
bfalse: ff
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
it: ⋅
, 
unit: Unit
, 
bool: 𝔹
, 
rev_implies: P 
⇐ Q
, 
iff: P 
⇐⇒ Q
, 
true: True
Lemmas referenced : 
geo-general-position_wf, 
list_wf, 
set_wf, 
Error :basic-geo-primitives_wf, 
Error :basic-geo-structure_wf, 
basic-geometry-_wf, 
Error :oriented-plane_wf, 
subtype_rel_transitivity, 
Error :oriented-plane-subtype, 
basic-geometry--subtype, 
geo-point_wf, 
length_wf, 
int_seg_wf, 
equal_wf, 
not_wf, 
int_formula_prop_less_lemma, 
intformless_wf, 
decidable__lt, 
int_formula_prop_wf, 
int_term_value_var_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_and_lemma, 
itermVar_wf, 
itermConstant_wf, 
intformle_wf, 
intformnot_wf, 
intformand_wf, 
full-omega-unsat, 
decidable__le, 
int_seg_properties, 
select_wf, 
Error :sq_stable__geo-lsep, 
lelt_wf, 
int_formula_prop_eq_lemma, 
intformeq_wf, 
nat_properties, 
nat_wf, 
false_wf, 
int_seg_subtype_nat, 
imax_nat, 
imax_wf, 
imax_strict_lb, 
le_wf, 
assert-bnot, 
bool_subtype_base, 
bool_cases_sqequal, 
eqff_to_assert, 
assert_of_le_int, 
eqtt_to_assert, 
bool_wf, 
le_int_wf, 
iff_weakening_equal, 
imax_unfold, 
true_wf, 
squash_wf, 
int_subtype_base, 
subtype_base_sq, 
lsep-all-sym
Rules used in proof : 
lambdaEquality, 
because_Cache, 
sqequalRule, 
independent_isectElimination, 
instantiate, 
applyEquality, 
natural_numberEquality, 
hypothesis, 
hypothesisEquality, 
rename, 
setElimination, 
intEquality, 
thin, 
isectElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
cut, 
lambdaFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution, 
baseClosed, 
imageMemberEquality, 
imageElimination, 
independent_pairFormation, 
voidEquality, 
voidElimination, 
isect_memberEquality, 
int_eqEquality, 
dependent_pairFormation, 
independent_functionElimination, 
approximateComputation, 
unionElimination, 
productElimination, 
dependent_functionElimination, 
applyLambdaEquality, 
equalitySymmetry, 
equalityTransitivity, 
dependent_set_memberEquality, 
promote_hyp, 
equalityElimination, 
universeEquality, 
cumulativity
Latex:
\mforall{}g:OrientedPlane.  \mforall{}xs:\{xs:Point  List|  geo-general-position(g;xs)\}  .  \mforall{}i,j,k:\mBbbN{}||xs||.
    ((\mneg{}(i  =  j))  {}\mRightarrow{}  (\mneg{}(k  =  i))  {}\mRightarrow{}  (\mneg{}(k  =  j))  {}\mRightarrow{}  xs[i]  \#  xs[j]xs[k])
Date html generated:
2017_10_02-PM-06_50_48
Last ObjectModification:
2017_08_06-PM-07_30_40
Theory : euclidean!plane!geometry
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