Nuprl Lemma : real-ball-slice_wf
∀[r:{r:ℝ| r0 ≤ r} ]
∀n:ℕ+. ∀t:{t:ℝ| t ∈ [-(r), r]} . ∀i:ℕn. ∀p:B(n - 1;ball-slice-radius(r;t)). (real-ball-slice(p;i;t) ∈ B(n;r))
Proof
Definitions occuring in Statement :
real-ball-slice: real-ball-slice(p;i;t),
ball-slice-radius: ball-slice-radius(r;t),
real-ball: B(n;r),
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
real-vec: ℝ^n,
int_seg: {i..j-},
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
real-ball: B(n;r),
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
real-ball-slice: real-ball-slice(p;i;t),
nat: ℕ,
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
req-vec: req-vec(n;x;y),
cand: A c∧ B,
subtract: n - m,
true: True,
nequal: a ≠ b ∈ T ,
dot-product: x⋅y,
so_lambda: λ2x.t[x],
so_apply: x[s],
eq_int: (i =z j),
req_int_terms: t1 ≡ t2
Lemmas referenced :
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
int_seg_properties,
nat_plus_properties,
decidable__lt,
subtract_wf,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
member_rccint_lemma,
square-rleq-implies,
real-vec-norm_wf,
sq_stable__rleq,
int-to-real_wf,
rleq_wf,
real-ball_wf,
ball-slice-radius_wf,
int_seg_wf,
real_wf,
i-member_wf,
rccint_wf,
rminus_wf,
nat_plus_wf,
rnexp_wf,
dot-product_wf,
nat_plus_subtype_nat,
real-vec-norm-nonneg,
rleq_weakening_equal,
radd_wf,
rleq_weakening,
rleq_functionality,
real-vec-norm-squared,
req_weakening,
rleq_functionality_wrt_implies,
rnexp_functionality_wrt_rleq,
radd_functionality_wrt_rleq,
dot-product-split,
add-member-int_seg1,
itermAdd_wf,
int_term_value_add_lemma,
req_functionality,
real-vec-subtype,
subtype_rel_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
minus-one-mul,
minus-add,
minus-minus,
add-associates,
add-swap,
minus-one-mul-top,
add-commutes,
zero-add,
sq_stable__le,
less-iff-le,
add_functionality_wrt_le,
le-add-cancel,
subtype_rel_self,
int_seg_subtype_nat,
radd_functionality,
dot-product_functionality,
eq_int_wf,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
neg_assert_of_eq_int,
decidable__equal_int,
int_subtype_base,
trivial-int-eq1,
rsum_wf,
rmul_wf,
equal-wf-base,
rmul_comm,
dot-product-comm,
rsum-single,
rnexp2,
radd-preserves-req,
itermMinus_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
dot-product-split-first,
ifthenelse_wf,
radd_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
functionExtensionality,
sqequalRule,
sqequalHypSubstitution,
setElimination,
thin,
rename,
because_Cache,
hypothesis,
extract_by_obid,
isectElimination,
inhabitedIsType,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
applyEquality,
dependent_set_memberEquality_alt,
hypothesisEquality,
independent_pairFormation,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
productIsType,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
imageMemberEquality,
baseClosed,
setIsType,
axiomEquality,
functionIsTypeImplies,
addEquality,
closedConclusion,
minusEquality,
intEquality,
sqequalBase
Latex:
\mforall{}[r:\{r:\mBbbR{}| r0 \mleq{} r\} ]
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}t:\{t:\mBbbR{}| t \mmember{} [-(r), r]\} . \mforall{}i:\mBbbN{}n. \mforall{}p:B(n - 1;ball-slice-radius(r;t)).
(real-ball-slice(p;i;t) \mmember{} B(n;r))
Date html generated:
2019_10_30-AM-10_15_06
Last ObjectModification:
2019_06_28-PM-01_52_15
Theory : real!vectors
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