Nuprl Lemma : in-hull-unique-next
∀g:OrientedPlane. ∀xs:{xs:Point List| geo-general-position(g;xs)} . ∀i,j:ℕ||xs||.
((¬(i = j ∈ ℤ)) ⇒ ij ∈ Hull(xs) ⇒ (∃!k:ℕ||xs||. ((¬(k = i ∈ 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-},
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},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
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,
not: ¬A,
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,
Error :geo-primitives_wf,
Error :geo-structure_wf,
Error :oriented-plane_wf,
subtype_rel_transitivity,
Error :oriented-plane_subtype,
Error :real-geometry-subtype,
Error :geo-structure-subtype-primitives,
all_wf,
Error :geo-point_wf,
length_wf,
int_seg_wf,
geo-general-position_wf,
in-hull_wf,
equal_wf,
not_wf,
in-hull-next,
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,
satisfiable-full-omega-tt,
decidable__equal_int,
int_seg_properties,
in-hull-unique2
Rules used in proof :
functionEquality,
lambdaEquality,
instantiate,
natural_numberEquality,
independent_isectElimination,
sqequalRule,
because_Cache,
applyEquality,
dependent_set_memberEquality,
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,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
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) {}\mRightarrow{} (\mexists{}!k:\mBbbN{}||xs||. ((\mneg{}(k = i)) \mwedge{} ki \mmember{} Hull(xs))))
Date html generated:
2017_10_02-PM-06_53_49
Last ObjectModification:
2017_08_06-PM-07_32_33
Theory : euclidean!plane!geometry
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