Nuprl Lemma : geo-tar-same-side-iff
∀e:BasicGeometry. ∀A,B,P,Q:Point. (geo-tar-same-side(e;P;Q;A;B) ⇐⇒ P # AB ∧ Q # AB ∧ (P leftof AB ⇐⇒ Q leftof AB))
Proof
Definitions occuring in Statement :
geo-tar-same-side: geo-tar-same-side(e;a;b;p;q),
basic-geometry: BasicGeometry,
geo-lsep: a # bc,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
geo-eq: a ≡ b,
subtract: n - m,
cons: [a / b],
select: L[n],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
geo-midpoint: a=m=b,
basic-geometry: BasicGeometry,
not: ¬A,
false: False,
geo-lsep: a # bc,
or: P ∨ Q,
geo-tar-same-side: geo-tar-same-side(e;a;b;p;q),
exists: ∃x:A. B[x],
geo-tar-opp-side: geo-tar-opp-side(e;a;b;p;q),
cand: A c∧ B,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-tar-opp-side_wf,
geo-tar-opp-side-iff,
geo-eq-self,
istype-less_than,
istype-le,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
intformle_wf,
intformnot_wf,
full-omega-unsat,
decidable__le,
length_of_nil_lemma,
istype-void,
length_of_cons_lemma,
geo-between-implies-colinear,
geo-colinear-is-colinear-set,
geo-congruent-sep,
geo-congruent-symmetry,
lsep-all-sym,
colinear-lsep-cycle,
left-implies-sep,
geo-sep-sym,
symmetric-point-construction,
not-left-and-right,
not-lsep-iff-colinear,
left-between,
euclidean-plane-subtype-oriented,
oriented-plane_wf,
geo-between_wf,
geo-between-symmetry,
lsep-all-sym2,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
geo-left_wf,
geo-lsep_wf,
geo-tar-same-side_wf
Rules used in proof :
lambdaEquality_alt,
approximateComputation,
natural_numberEquality,
dependent_set_memberEquality_alt,
isect_memberEquality_alt,
dependent_pairFormation_alt,
rename,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
unionElimination,
productElimination,
independent_isectElimination,
instantiate,
inhabitedIsType,
functionIsType,
because_Cache,
applyEquality,
productIsType,
sqequalRule,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
universeIsType,
independent_pairFormation,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}A,B,P,Q:Point.
(geo-tar-same-side(e;P;Q;A;B) \mLeftarrow{}{}\mRightarrow{} P \# AB \mwedge{} Q \# AB \mwedge{} (P leftof AB \mLeftarrow{}{}\mRightarrow{} Q leftof AB))
Date html generated:
2019_10_29-AM-09_15_10
Last ObjectModification:
2019_10_18-PM-03_17_15
Theory : euclidean!plane!geometry
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