Nuprl Lemma : left-test-symmetry
∀g:OrientedPlane. ∀xs:{xs:Point List| geo-general-position(g;xs)} . ∀i:ℕ||xs||. ∀j:{j:ℕ||xs||| ¬(i = j ∈ ℤ)} .
∀k:{k:ℕ||xs||| (¬(k = i ∈ ℤ)) ∧ (¬(k = j ∈ ℤ))} .
i L jk = j L ki
Proof
Definitions occuring in Statement :
left-test: i L jk,
geo-general-position: geo-general-position(g;xs),
oriented-plane: OrientedPlane,
geo-point: Point,
length: ||as||,
list: T List,
int_seg: {i..j-},
bool: 𝔹,
all: ∀x:A. B[x],
not: ¬A,
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],
int_seg: {i..j-},
and: P ∧ Q,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
left-test: i L jk,
all: ∀x:A. B[x],
oriented-plane: Error :oriented-plane,
less_than: a < b,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
squash: ↓T,
sq_stable: SqStable(P),
false: False,
not: ¬A,
implies: P ⇒ Q,
lelt: i ≤ j < k
Lemmas referenced :
geo-general-position_wf,
list_wf,
equal_wf,
not_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry-_wf,
Error :oriented-plane_wf,
subtype_rel_transitivity,
Error :oriented-plane-subtype,
basic-geometry--subtype,
geo-point_wf,
length_wf,
int_seg_wf,
set_wf,
geo-lsep_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
int_seg_properties,
squash_wf,
sq_stable__not,
sq_stable__and,
select_wf,
isleft-symmetry,
lelt_wf,
geo-general-position-implies
Rules used in proof :
intEquality,
productEquality,
lambdaEquality,
because_Cache,
rename,
setElimination,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesisEquality,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
hypothesis,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
voidEquality,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
unionElimination,
imageElimination,
baseClosed,
imageMemberEquality,
productElimination,
voidElimination,
independent_functionElimination,
dependent_set_memberEquality,
dependent_functionElimination
Latex:
\mforall{}g:OrientedPlane. \mforall{}xs:\{xs:Point List| geo-general-position(g;xs)\} . \mforall{}i:\mBbbN{}||xs||. \mforall{}j:\{j:\mBbbN{}||xs|||
\mneg{}(i = j)\} .
\mforall{}k:\{k:\mBbbN{}||xs||| (\mneg{}(k = i)) \mwedge{} (\mneg{}(k = j))\} .
i L jk = j L ki
Date html generated:
2017_10_02-PM-06_51_20
Last ObjectModification:
2017_08_08-PM-00_39_14
Theory : euclidean!plane!geometry
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