Nuprl Lemma : m-regularize-mcauchy
∀[X:Type]. ∀[d:metric(X)]. ∀[s:ℕ ⟶ X]. (λk.(6 * k) ∈ mcauchy(d;n.m-regularize(d;s) n))
Proof
Definitions occuring in Statement :
m-regularize: m-regularize(d;s),
mcauchy: mcauchy(d;n.x[n]),
metric: metric(X),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
multiply: n * m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
mcauchy: mcauchy(d;n.x[n]),
member: t ∈ T,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
nat: ℕ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
m-regularize: m-regularize(d;s),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
has-value: (a)↓,
so_lambda: λ2x.t[x],
so_apply: x[s],
let: let,
sq_type: SQType(T),
bool: 𝔹,
m-not-reg: m-not-reg(d;s;n),
isl: isl(x),
m-reg-test: m-reg-test(d;b;s;x),
int-seg-case: int-seg-case(i;j;d),
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
subtract: n - m,
bfalse: ff,
uiff: uiff(P;Q),
unit: Unit,
it: ⋅,
btrue: tt,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
req_int_terms: t1 ≡ t2
Lemmas referenced :
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
intformless_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_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
rleq_wf,
mdist_wf,
m-regularize_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
decidable__lt,
rless_wf,
nat_plus_wf,
istype-nat,
metric_wf,
istype-universe,
first-m-not-reg-property,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
subtype_rel_function,
nat_wf,
int_seg_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
first-m-not-reg_wf,
value-type-has-value,
set-value-type,
lelt_wf,
int-value-type,
set_subtype_base,
int_subtype_base,
bool_wf,
m-not-reg_wf,
bfalse_wf,
istype-less_than,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__equal_int,
subtype_base_sq,
it_wf,
unit_wf2,
btrue_neq_bfalse,
lt_int_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
rleq-int-fractions2,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
rleq_functionality,
mdist-same,
req_weakening,
equal-wf-base,
le_int_wf,
le_wf,
bnot_wf,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
not-m-not-reg-3regular,
radd_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq-int-fractions,
mul_bounds_1b,
mul_nat_plus,
radd_functionality_wrt_rleq,
radd-int-fractions,
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,
mdist-symm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaEquality_alt,
dependent_set_memberEquality_alt,
multiplyEquality,
natural_numberEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
hypothesisEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
functionIsType,
because_Cache,
applyEquality,
closedConclusion,
inrFormation_alt,
productElimination,
instantiate,
universeEquality,
lambdaFormation_alt,
addEquality,
imageElimination,
inhabitedIsType,
callbyvalueReduce,
intEquality,
productIsType,
equalityIstype,
sqequalBase,
equalitySymmetry,
isectIsType,
baseClosed,
equalityTransitivity,
cumulativity,
inrEquality_alt,
equalityElimination,
promote_hyp,
baseApply
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[s:\mBbbN{} {}\mrightarrow{} X]. (\mlambda{}k.(6 * k) \mmember{} mcauchy(d;n.m-regularize(d;s) n))
Date html generated:
2019_10_30-AM-07_04_08
Last ObjectModification:
2019_10_09-AM-09_17_15
Theory : reals
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