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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