Nuprl Lemma : in-hull-next2
∀g:OrientedPlane. ∀xs:{xs:Point List| geo-general-position(g;xs)} . ∀i,j:ℕ||xs||.
((¬(i = j ∈ ℤ)) ⇒ (ij ∈ Hull(xs) ∧ 2 < ||xs||) ⇒ (∃k:ℕ||xs||. (((¬(k = i ∈ ℤ)) ∧ (¬(k = j ∈ ℤ))) ∧ ki ∈ Hull(xs))))
Proof
Definitions occuring in Statement :
in-hull: ij ∈ Hull(xs),
geo-general-position: geo-general-position(g;xs),
oriented-plane: OrientedPlane,
geo-point: Point,
length: ||as||,
list: T List,
int_seg: {i..j-},
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: 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],
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
cand: A c∧ B,
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
exists: ∃x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
in-hull: ij ∈ Hull(xs),
or: P ∨ Q,
decidable: Dec(P),
sq_type: SQType(T),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q)
Lemmas referenced :
list_wf,
set_wf,
int_seg_wf,
less_than_wf,
geo-general-position_wf,
in-hull_wf,
not_wf,
equal_wf,
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
int_seg_properties,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
oriented-plane_wf,
subtype_rel_transitivity,
oriented-plane-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
length_wf,
lelt_wf,
in-hull-leftmost,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
left-test-symmetry,
left-test_wf,
assert_functionality_wrt_uiff
Rules used in proof :
productEquality,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
approximateComputation,
independent_pairFormation,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
natural_numberEquality,
isectElimination,
dependent_set_memberEquality,
rename,
setElimination,
dependent_pairFormation,
hypothesis,
independent_functionElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
unionElimination,
equalitySymmetry,
equalityTransitivity,
cumulativity
Latex:
\mforall{}g:OrientedPlane. \mforall{}xs:\{xs:Point List| geo-general-position(g;xs)\} . \mforall{}i,j:\mBbbN{}||xs||.
((\mneg{}(i = j))
{}\mRightarrow{} (ij \mmember{} Hull(xs) \mwedge{} 2 < ||xs||)
{}\mRightarrow{} (\mexists{}k:\mBbbN{}||xs||. (((\mneg{}(k = i)) \mwedge{} (\mneg{}(k = j))) \mwedge{} ki \mmember{} Hull(xs))))
Date html generated:
2018_05_22-PM-00_07_27
Last ObjectModification:
2017_12_12-PM-04_23_39
Theory : euclidean!plane!geometry
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