Nuprl Lemma : in-hull-unique1
∀[g:OrientedPlane]. ∀[xs:{xs:Point List| geo-general-position(g;xs)} ]. ∀[i,j,k:ℕ||xs||].
(j = k ∈ ℤ) supposing (ik ∈ Hull(xs) and (¬(i = k ∈ ℤ)) and ij ∈ Hull(xs) and (¬(i = j ∈ ℤ)))
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-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
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],
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
and: P ∧ Q,
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
guard: {T},
not: ¬A,
implies: P ⇒ Q,
in-hull: ij ∈ Hull(xs),
or: P ∨ Q,
decidable: Dec(P),
int_seg: {i..j-},
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
cand: A c∧ B,
btrue: tt,
ifthenelse: if b then t else f fi ,
bnot: ¬bb
Lemmas referenced :
list_wf,
set_wf,
int_seg_wf,
not_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,
geo-general-position_wf,
in-hull_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,
satisfiable-full-omega-tt,
Error :geo-point_wf,
length_wf,
int_seg_properties,
decidable__equal_int,
left-test-symmetry,
left-test_wf,
assert_functionality_wrt_uiff,
bnot-left-test,
bnot_wf,
btrue_neq_bfalse,
bool_wf,
and_wf,
bfalse_wf,
assert_elim,
assert_of_bnot
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
axiomEquality,
instantiate,
dependent_set_memberEquality,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
productElimination,
sqequalRule,
applyEquality,
natural_numberEquality,
isectElimination,
lambdaFormation,
independent_functionElimination,
hypothesisEquality,
unionElimination,
hypothesis,
because_Cache,
rename,
setElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productEquality,
applyLambdaEquality
Latex:
\mforall{}[g:OrientedPlane]. \mforall{}[xs:\{xs:Point List| geo-general-position(g;xs)\} ]. \mforall{}[i,j,k:\mBbbN{}||xs||].
(j = k) supposing (ik \mmember{} Hull(xs) and (\mneg{}(i = k)) and ij \mmember{} Hull(xs) and (\mneg{}(i = j)))
Date html generated:
2017_10_02-PM-06_51_48
Last ObjectModification:
2017_08_06-PM-07_31_33
Theory : euclidean!plane!geometry
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