Nuprl Lemma : real-cube-uniform-continuity
∀k,n:ℕ. ∀a,b:ℕn ⟶ ℝ.
((∀i:ℕn. ((a i) < (b i)))
⇒ (∀f:{f:real-cube(n;a;b) ⟶ ℝ^k| ∀x,y:real-cube(n;a;b). (req-vec(n;x;y) ⇒ req-vec(k;f x;f y))} . ∀e:{e:ℝ| r0 < e} \000C.
∃d:ℕ+. ∀x,y:real-cube(n;a;b). ((d(x;y) ≤ (r1/r(d))) ⇒ (d(f x;f y) ≤ e))))
Proof
Definitions occuring in Statement :
real-cube: real-cube(n;a;b),
real-vec-dist: d(x;y),
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
false: False,
real-cube: real-cube(n;a;b),
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
rless: x < y,
sq_exists: ∃x:A [B[x]],
real-vec: ℝ^n,
less_than: a < b,
squash: ↓T,
true: True,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
sq_type: SQType(T),
cand: A c∧ B,
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
rfun: I ⟶ℝ,
real-fun: real-fun(f;a;b),
req-vec: req-vec(n;x;y),
real-cont: real-cont(f;a;b),
rleq: x ≤ y,
rnonneg: rnonneg(x),
rdiv: (x/y),
pi1: fst(t),
subtract: n - m,
real: ℝ,
i-member: r ∈ I,
rccint: [l, u],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
real-vec-dist: d(x;y),
real-vec-sub: X - Y,
dot-product: x⋅y,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
eq_int: (i =z j),
nequal: a ≠ b ∈ T ,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
label: ...$L... t,
int_nzero: ℤ-o
Lemmas referenced :
real_wf,
rless_wf,
int-to-real_wf,
real-cube_wf,
istype-void,
istype-le,
real-vec_wf,
req-vec_wf,
int_seg_wf,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
nat_plus_wf,
rleq_wf,
real-vec-dist_wf,
nat_plus_properties,
rdiv_wf,
rless-int,
decidable__lt,
istype-less_than,
primrec-wf2,
all_wf,
exists_wf,
istype-nat,
istype-top,
sq_stable__rless,
rleq_weakening_rless,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
rleq_functionality,
real-vec-dist-same-zero,
req_weakening,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
member_rccint_lemma,
sq_stable__rleq,
intformeq_wf,
int_formula_prop_eq_lemma,
int_seg_subtype_special,
int_seg_cases,
i-member_wf,
rccint_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
sq_stable__req,
req_wf,
small-reciprocal-real,
rmul_preserves_rless,
rless_transitivity2,
rabs_wf,
rsub_wf,
rmul_wf,
itermMultiply_wf,
rinv_wf2,
le_witness_for_triv,
rless_functionality,
req_transitivity,
rmul-rinv3,
real_term_value_mul_lemma,
subtype_rel_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
rmin_wf,
rmin_strict_ub,
sq_stable__less_than,
rmin-rleq,
set_subtype_base,
lelt_wf,
real-vec-dist-dim1,
sq_stable__i-member,
nat_wf,
le_wf,
real-vec-dist-dim0,
implies-real-vec-dist-rleq,
rsqrt_wf,
rleq-int,
rmul_preserves_rleq,
rsqrt-rleq-iff,
rnexp_wf,
rnexp2,
rmul-int,
mul_preserves_le,
int_term_value_mul_lemma,
rmul_preserves_rleq2,
ifthenelse_wf,
lt_int_wf,
itermAdd_wf,
int_term_value_add_lemma,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
iff_imp_equal_bool,
btrue_wf,
iff_functionality_wrt_iff,
istype-true,
subtract-add-cancel,
bnot_wf,
not_wf,
istype-assert,
zero-add,
ite_rw_false,
bool_cases,
iff_transitivity,
assert_of_bnot,
add-member-int_seg2,
square-rleq-implies,
real-vec-dist-nonneg,
rmul-rinv,
real-vec-norm_wf,
real-vec-sub_wf,
dot-product_wf,
real-vec-norm-squared,
dot-product_functionality,
rsum_functionality,
rmul_comm,
rsum_wf,
rsum-shift,
rsum-split-first,
radd_wf,
square-nonneg,
trivial-rleq-radd,
radd_functionality_wrt_rleq,
rsum-split2,
req_inversion,
i-member_functionality,
real-vec-dist_functionality,
rneq-int,
not_functionality_wrt_implies,
equal-wf-base,
rationals_wf,
equal_functionality_wrt_subtype_rel2,
int-subtype-rationals,
int_nzero-rational,
nat_plus_inc_int_nzero,
proper-interval-to-int-bounded,
absval_pos,
nat_plus_subtype_nat,
rleq-int-fractions,
imax_wf,
imax_nat_plus,
imax_ub,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
rsum_functionality_wrt_rleq,
real-vec-triangle-inequality,
rleq_transitivity,
radd-rdiv,
nequal_wf,
int-rinv-cancel,
real_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
setIsType,
universeIsType,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
hypothesisEquality,
functionIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
sqequalRule,
voidElimination,
because_Cache,
setElimination,
rename,
applyEquality,
productElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
inhabitedIsType,
unionElimination,
productIsType,
closedConclusion,
inrFormation_alt,
equalityTransitivity,
equalitySymmetry,
functionEquality,
setEquality,
functionExtensionality,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
universeEquality,
cumulativity,
intEquality,
hypothesis_subsumption,
productEquality,
promote_hyp,
functionIsTypeImplies,
equalityIstype,
addEquality,
minusEquality,
multiplyEquality,
hyp_replacement,
applyLambdaEquality,
equalityElimination,
inlFormation_alt,
sqequalBase
Latex:
\mforall{}k,n:\mBbbN{}. \mforall{}a,b:\mBbbN{}n {}\mrightarrow{} \mBbbR{}.
((\mforall{}i:\mBbbN{}n. ((a i) < (b i)))
{}\mRightarrow{} (\mforall{}f:\{f:real-cube(n;a;b) {}\mrightarrow{} \mBbbR{}\^{}k|
\mforall{}x,y:real-cube(n;a;b). (req-vec(n;x;y) {}\mRightarrow{} req-vec(k;f x;f y))\} . \mforall{}e:\{e:\mBbbR{}| r0 < e\} .
\mexists{}d:\mBbbN{}\msupplus{}. \mforall{}x,y:real-cube(n;a;b). ((d(x;y) \mleq{} (r1/r(d))) {}\mRightarrow{} (d(f x;f y) \mleq{} e))))
Date html generated:
2019_10_30-AM-10_14_34
Last ObjectModification:
2019_06_28-PM-01_52_02
Theory : real!vectors
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