Nuprl Lemma : 1-dim-cube-endpoints
∀k:ℕ. ∀c:ℚCube(k).
((dim(c) = 1 ∈ ℤ)
⇒ (∃a,b:ℚCube(k)
(a ≤ c
∧ b ≤ c
∧ (dim(a) = 0 ∈ ℤ)
∧ (dim(b) = 0 ∈ ℤ)
∧ (∀p,q:ℝ^k. ((¬¬in-rat-cube(k;p;a)) ⇒ (¬¬in-rat-cube(k;q;b)) ⇒ p ≠ q))
∧ (∀f:ℚCube(k). (f ≤ c ⇒ (dim(f) = 0 ∈ ℤ) ⇒ ((f = a ∈ ℚCube(k)) ∨ (f = b ∈ ℚCube(k))))))))
Proof
Definitions occuring in Statement :
in-rat-cube: in-rat-cube(k;p;c),
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
rat-cube-dimension: dim(c),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k)
Definitions unfolded in proof :
concat: concat(ll),
less_than': less_than'(a;b),
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
nat_plus: ℕ+,
lt_int: i <z j,
from-upto: [n, m),
upto: upto(n),
sq_stable: SqStable(P),
mapfilter: mapfilter(f;P;L),
rat-cube-faces: rat-cube-faces(k;c),
pi2: snd(t),
rat-interval-dimension: dim(I),
rat-point-interval: [a],
lower-rc-face: lower-rc-face(c;j),
upper-rc-face: upper-rc-face(c;j),
rev_uimplies: rev_uimplies(P;Q),
pi1: fst(t),
rational-interval: ℚInterval,
real-vec: ℝ^n,
so_apply: x[s],
so_lambda: λ2x.t[x],
decidable: Dec(P),
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
nat: ℕ,
less_than: a < b,
it: ⋅,
unit: Unit,
bool: 𝔹,
rational-cube: ℚCube(k),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
prop: ℙ,
squash: ↓T,
cand: A c∧ B,
exists: ∃x:A. B[x],
false: False,
true: True,
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
and: P ∧ Q,
uiff: uiff(P;Q),
guard: {T},
sq_type: SQType(T),
uimplies: b supposing a,
or: P ∨ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
rat-cube-dimension: dim(c),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
member_singleton,
cons_member,
reduce_nil_lemma,
reduce_cons_lemma,
map_nil_lemma,
map_cons_lemma,
istype-false,
int_seg_subtype_nat,
list_ind_nil_lemma,
list_ind_cons_lemma,
filter_cons_lemma,
filter_append,
nil_wf,
cons_wf,
subtype_rel_list,
subtract_wf,
append_wf,
upto_decomp1,
list_subtype_base,
list_wf,
subtract-1-ge-0,
less_than_wf,
assert_wf,
iff_weakening_uiff,
filter_nil_lemma,
assert_of_lt_int,
lt_int_wf,
ge_wf,
decidable__equal_int_seg,
sq_stable__l_member,
istype-assert,
l_member_wf,
istype-less_than,
istype-le,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
upto_wf,
filter-sq,
member-rat-cube-faces,
qless_wf,
qless_irreflexivity,
qle_weakening_eq_qorder,
qless_transitivity_2_qorder,
assert-q_less-eq,
q_less_wf,
rneq_wf,
rneq-rat2real,
real-vec-sep-iff-rneq,
rational-interval_wf,
int_seg_wf,
rat2real_wf,
real-vec-sep_functionality,
not-not-in-0-dim-cube,
istype-nat,
rational-cube_wf,
real-vec-sep_wf,
rat-cube-face_wf,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
real-vec_wf,
in-rat-cube_wf,
upper-rc-face-dimension,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
decidable__equal_int,
neg_assert_of_eq_int,
assert-bnot,
bool_cases_sqequal,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
int_seg_properties,
assert_of_eq_int,
rat-interval-dimension_wf,
eq_int_wf,
iff_weakening_equal,
subtype_rel_self,
lower-rc-face-dimension,
istype-universe,
true_wf,
squash_wf,
equal_wf,
upper-rc-face-is-face,
lower-rc-face-is-face,
upper-rc-face_wf,
lower-rc-face_wf,
rat-cube-dimension-1,
int_subtype_base,
assert_of_bnot,
eqff_to_assert,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
inhabited-rat-cube_wf
Rules used in proof :
inrFormation_alt,
inlFormation_alt,
functionIsTypeImplies,
axiomSqEquality,
intWeakElimination,
setEquality,
applyLambdaEquality,
setIsType,
dependent_set_memberEquality_alt,
functionEquality,
unionIsType,
productIsType,
sqequalBase,
addEquality,
minusEquality,
functionIsType,
promote_hyp,
equalityIstype,
isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
equalityElimination,
baseClosed,
imageMemberEquality,
rename,
setElimination,
universeEquality,
inhabitedIsType,
universeIsType,
imageElimination,
lambdaEquality_alt,
applyEquality,
independent_pairFormation,
dependent_pairFormation_alt,
voidElimination,
natural_numberEquality,
intEquality,
sqequalRule,
productElimination,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
cumulativity,
instantiate,
unionElimination,
because_Cache,
dependent_functionElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
introduction,
sqequalHypSubstitution,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c:\mBbbQ{}Cube(k).
((dim(c) = 1)
{}\mRightarrow{} (\mexists{}a,b:\mBbbQ{}Cube(k)
(a \mleq{} c
\mwedge{} b \mleq{} c
\mwedge{} (dim(a) = 0)
\mwedge{} (dim(b) = 0)
\mwedge{} (\mforall{}p,q:\mBbbR{}\^{}k. ((\mneg{}\mneg{}in-rat-cube(k;p;a)) {}\mRightarrow{} (\mneg{}\mneg{}in-rat-cube(k;q;b)) {}\mRightarrow{} p \mneq{} q))
\mwedge{} (\mforall{}f:\mBbbQ{}Cube(k). (f \mleq{} c {}\mRightarrow{} (dim(f) = 0) {}\mRightarrow{} ((f = a) \mvee{} (f = b)))))))
Date html generated:
2019_10_30-AM-10_13_32
Last ObjectModification:
2019_10_28-PM-04_51_36
Theory : real!vectors
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