Nuprl Lemma : hull-cmp_wf
∀[g:OrientedPlane]. ∀[xs:{xs:Point List| geo-general-position(g;xs)} ]. ∀[i,j:ℕ||xs||].
(hull-cmp(g;xs;i;j) ∈ comparison({k:ℕ||xs||| (¬(k = i ∈ ℤ)) ∧ (¬(k = j ∈ ℤ))} )) supposing (ij ∈ Hull(xs) and (¬(i = j\000C ∈ ℤ)))
Proof
Definitions occuring in Statement :
hull-cmp: hull-cmp(g;xs;i;j),
in-hull: ij ∈ Hull(xs),
geo-general-position: geo-general-position(g;xs),
oriented-plane: OrientedPlane,
geo-point: Point,
comparison: comparison(T),
length: ||as||,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
comparison: comparison(T),
hull-cmp: hull-cmp(g;xs;i;j),
and: P ∧ Q,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
nequal: a ≠ b ∈ T ,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
true: True,
trans: Trans(T;x,y.E[x; y]),
sq_stable: SqStable(P),
less_than': less_than'(a;b)
Latex:
\mforall{}[g:OrientedPlane]. \mforall{}[xs:\{xs:Point List| geo-general-position(g;xs)\} ]. \mforall{}[i,j:\mBbbN{}||xs||].
(hull-cmp(g;xs;i;j) \mmember{} comparison(\{k:\mBbbN{}||xs||| (\mneg{}(k = i)) \mwedge{} (\mneg{}(k = j))\} )) supposing (ij \mmember{} Hull(xs) \000Cand (\mneg{}(i = j)))
Date html generated:
2020_05_20-AM-10_02_35
Last ObjectModification:
2019_12_30-PM-01_14_32
Theory : euclidean!plane!geometry
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