Nuprl Lemma : in-hull-unique-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,
exists!: ∃!x:T. P[x],
all: ∀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],
guard: {T},
not: ¬A,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
prop: ℙ,
cand: A c∧ B,
and: P ∧ Q,
exists: ∃x:A. B[x],
exists!: ∃!x:T. P[x],
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x],
squash: ↓T,
less_than: a < b,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k
Lemmas referenced :
list_wf,
set_wf,
less_than_wf,
geo-general-position_wf,
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,
all_wf,
geo-point_wf,
length_wf,
int_seg_wf,
in-hull_wf,
equal_wf,
not_wf,
in-hull-next2,
lelt_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
itermConstant_wf,
intformle_wf,
decidable__le,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__equal_int,
int_seg_properties,
in-hull-unique2
Rules used in proof :
functionEquality,
lambdaEquality,
instantiate,
sqequalRule,
applyEquality,
natural_numberEquality,
independent_isectElimination,
dependent_set_memberEquality,
because_Cache,
rename,
setElimination,
intEquality,
isectElimination,
productEquality,
independent_pairFormation,
dependent_pairFormation,
productElimination,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
imageElimination,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
approximateComputation,
unionElimination
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_35
Last ObjectModification:
2017_12_12-PM-04_26_44
Theory : euclidean!plane!geometry
Home
Index