Nuprl Lemma : not-m-not-reg-3regular
∀[X:Type]. ∀[d:metric(X)]. ∀[s:ℕ ⟶ X]. ∀[b:ℕ].
((∀n:ℕb. m-not-reg(d;s;n) = ff) ⇒ (∀n,m:ℕb. (mdist(d;s n;s m) ≤ ((r(3)/r(n + 1)) + (r(3)/r(m + 1))))))
Proof
Definitions occuring in Statement :
m-not-reg: m-not-reg(d;s;n),
mdist: mdist(d;x;y),
metric: metric(X),
rdiv: (x/y),
rleq: x ≤ y,
radd: a + b,
int-to-real: r(n),
int_seg: {i..j-},
nat: ℕ,
bfalse: ff,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
m-not-reg: m-not-reg(d;s;n),
nat: ℕ,
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
isl: isl(x),
rev_uimplies: rev_uimplies(P;Q),
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rge: x ≥ y,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
rleq: x ≤ y,
rnonneg: rnonneg(x),
uiff: uiff(P;Q),
nat_plus: ℕ+,
req_int_terms: t1 ≡ t2
Lemmas referenced :
decidable__lt,
m-reg-test_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
subtype_rel_function,
nat_wf,
int_seg_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
btrue_neq_bfalse,
rleq_functionality_wrt_implies,
mdist_wf,
radd_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
rless_wf,
rleq_weakening_rless,
istype-less_than,
rleq_weakening_equal,
rleq_weakening,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
bool_wf,
m-not-reg_wf,
bfalse_wf,
le_witness_for_triv,
istype-nat,
metric_wf,
istype-universe,
itermSubtract_wf,
req-iff-rsub-is-0,
radd-non-neg,
rleq-int-fractions2,
itermMultiply_wf,
int_term_value_mul_lemma,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq_functionality,
mdist-symm,
req_weakening,
mdist-same
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
unionElimination,
isectElimination,
dependent_set_memberEquality_alt,
productElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
applyEquality,
because_Cache,
inhabitedIsType,
equalityIstype,
baseClosed,
sqequalBase,
equalitySymmetry,
closedConclusion,
addEquality,
inrFormation_alt,
productIsType,
equalityTransitivity,
instantiate,
cumulativity,
intEquality,
functionIsType,
functionIsTypeImplies,
isectIsTypeImplies,
universeEquality,
multiplyEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[s:\mBbbN{} {}\mrightarrow{} X]. \mforall{}[b:\mBbbN{}].
((\mforall{}n:\mBbbN{}b. m-not-reg(d;s;n) = ff)
{}\mRightarrow{} (\mforall{}n,m:\mBbbN{}b. (mdist(d;s n;s m) \mleq{} ((r(3)/r(n + 1)) + (r(3)/r(m + 1))))))
Date html generated:
2019_10_30-AM-07_01_12
Last ObjectModification:
2019_10_09-AM-09_03_26
Theory : reals
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