Nuprl Lemma : in-hull-transitivity
∀g:OrientedPlane. ∀xs:{xs:Point List| geo-general-position(g;xs)} . ∀i,j:ℕ||xs||.
((¬(i = j ∈ ℤ)) ⇒ ij ∈ Hull(xs) ⇒ Trans({k:ℕ||xs||| (¬(k = i ∈ ℤ)) ∧ (¬(k = j ∈ ℤ))} ;x,y.(¬(x = y ∈ ℤ)) ∧ (↑x L iy)\000C))
Proof
Definitions occuring in Statement :
in-hull: ij ∈ Hull(xs),
left-test: i L jk,
geo-general-position: geo-general-position(g;xs),
oriented-plane: OrientedPlane,
geo-point: Point,
length: ||as||,
list: T List,
trans: Trans(T;x,y.E[x; y]),
int_seg: {i..j-},
assert: ↑b,
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],
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
guard: {T},
not: ¬A,
cand: A c∧ B,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
and: P ∧ Q,
prop: ℙ,
member: t ∈ T,
trans: Trans(T;x,y.E[x; y]),
implies: P ⇒ Q,
all: ∀x:A. B[x],
sq_type: SQType(T),
uiff: uiff(P;Q),
btrue: tt,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
in-hull: ij ∈ Hull(xs),
squash: ↓T,
sq_stable: SqStable(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
or: P ∨ Q,
decidable: Dec(P),
geo-lsep: a # bc
Lemmas referenced :
list_wf,
geo-general-position_wf,
in-hull_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
Error :o-geo-structure_wf,
Error :oriented-plane_wf,
subtype_rel_transitivity,
Error :oriented-plane-subtype1,
Error :o-geo-structure-subtype,
int_seg_wf,
set_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,
geo-point_wf,
length_wf,
int_seg_properties,
left-test_wf,
assert_wf,
equal_wf,
not_wf,
int_subtype_base,
subtype_base_sq,
left-test-symmetry,
satisfiable-full-omega-tt,
Error :geo-point_wf,
Error :geo-primitives_wf,
Error :geo-structure_wf,
Error :oriented-plane_wf,
Error :oriented-plane_subtype,
Error :real-geometry-subtype,
Error :geo-structure-subtype-primitives,
assert_functionality_wrt_uiff,
bnot-left-test,
bnot_wf,
btrue_neq_bfalse,
and_wf,
bfalse_wf,
assert_elim,
bool_wf,
lelt_wf,
sq_stable__assert,
assert-left-test,
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,
select_wf,
geo-left-transitivity,
geo-general-position-implies
Rules used in proof :
instantiate,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
sqequalRule,
applyEquality,
natural_numberEquality,
dependent_set_memberEquality,
hypothesisEquality,
hypothesis,
because_Cache,
isectElimination,
extract_by_obid,
introduction,
productEquality,
productElimination,
sqequalHypSubstitution,
cut,
rename,
thin,
setElimination,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
equalitySymmetry,
equalityTransitivity,
cumulativity,
promote_hyp,
computeAll,
applyLambdaEquality,
levelHypothesis,
addLevel,
imageElimination,
baseClosed,
imageMemberEquality,
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{} Trans(\{k:\mBbbN{}||xs||| (\mneg{}(k = i)) \mwedge{} (\mneg{}(k = j))\} ;x,y.(\mneg{}(x = y)) \mwedge{} (\muparrow{}x L\000C iy)))
Date html generated:
2017_10_02-PM-06_52_06
Last ObjectModification:
2017_08_08-PM-00_38_55
Theory : euclidean!plane!geometry
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