Nuprl Lemma : left-between-weak
∀g:OrientedPlane. ∀a,b,x,y,z:Point. (x leftof ab ⇒ y leftof ab ⇒ B(xzy) ⇒ (¬z leftof ba))
Proof
Definitions occuring in Statement :
oriented-plane: OrientedPlane,
geo-between: B(abc),
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
basic-geometry-: BasicGeometry-,
lelt: i ≤ j < k,
int_seg: {i..j-},
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
ge: i ≥ j ,
true: True,
squash: ↓T,
less_than: a < b,
cand: A c∧ B,
subtract: n - m,
cons: [a / b],
select: L[n],
top: Top,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
euclidean-plane: EuclideanPlane,
and: P ∧ Q,
iff: P ⇐⇒ Q,
geo-eq: a ≡ b,
stable: Stable{P},
oriented-plane: OrientedPlane,
or: P ∨ Q,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
false: False,
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
not-left-and-right,
geo-sep-irrefl',
geo-between-same,
basic-geometry-_wf,
subtype_rel_self,
geo-between-exchange3,
geo-between-inner-trans,
geo-between-symmetry,
left-not-colinear,
geo-sep-sym,
left-implies-sep,
geo-colinear_functionality,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
list_ind_nil_lemma,
list_ind_cons_lemma,
geo-between-implies-colinear,
geo-colinear-is-colinear-set,
l_member_wf,
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,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
select_wf,
length_wf,
istype-less_than,
length_of_nil_lemma,
length_of_cons_lemma,
istype-le,
geo-between-sep,
nil_wf,
cons_wf,
geo-colinear-append,
geo-sep_functionality,
use-plane-sep,
minimal-not-not-excluded-middle,
geo-between_functionality,
geo-eq_weakening,
geo-left_functionality,
minimal-double-negation-hyp-elim,
istype-void,
not_wf,
geo-sep_wf,
false_wf,
stable__false,
geo-point_wf,
geo-between_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
oriented-plane_wf,
subtype_rel_transitivity,
oriented-plane-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-left_wf
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
int_eqEquality,
lambdaEquality_alt,
approximateComputation,
equalityIstype,
productIsType,
baseClosed,
imageMemberEquality,
isect_memberEquality_alt,
natural_numberEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
dependent_pairFormation_alt,
rename,
setElimination,
productElimination,
dependent_functionElimination,
promote_hyp,
unionElimination,
unionIsType,
functionIsType,
functionEquality,
unionEquality,
inhabitedIsType,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
universeIsType,
voidElimination,
independent_functionElimination,
sqequalHypSubstitution,
hypothesis,
because_Cache,
thin,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:OrientedPlane. \mforall{}a,b,x,y,z:Point. (x leftof ab {}\mRightarrow{} y leftof ab {}\mRightarrow{} B(xzy) {}\mRightarrow{} (\mneg{}z leftof ba))
Date html generated:
2019_10_29-AM-09_13_29
Last ObjectModification:
2019_10_18-PM-03_17_00
Theory : euclidean!plane!geometry
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