Nuprl Lemma : case-real3-seq_wf
∀[f:ℕ+ ⟶ 𝔹]. ∀[b:ℝ]. ∀[a:ℝ supposing ∃n:ℕ+. (↑(f n))].
case-real3-seq(a;b;f) ∈ {s:ℕ+ ⟶ ℤ| 3-regular-seq(s)} supposing ∀n,m:ℕ+. ((↑(f n)) ⇒ (¬↑(f m)) ⇒ (|(a m) - b m| ≤ \000C4))
Proof
Definitions occuring in Statement :
case-real3-seq: case-real3-seq(a;b;f),
real: ℝ,
regular-int-seq: k-regular-seq(f),
absval: |i|,
nat_plus: ℕ+,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
case-real3-seq: case-real3-seq(a;b;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
prop: ℙ,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
assert: ↑b,
true: True,
real: ℝ,
bfalse: ff,
regular-int-seq: k-regular-seq(f),
uiff: uiff(P;Q),
and: P ∧ Q,
nat_plus: ℕ+,
nat: ℕ,
sq_stable: SqStable(P),
squash: ↓T,
or: P ∨ Q,
bnot: ¬bb,
false: False,
not: ¬A,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
uimplies_subtype,
nat_plus_wf,
assert_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
istype-assert,
eqtt_to_assert,
sq_stable__le,
absval_wf,
subtract_wf,
real_wf,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
regular-int-seq_wf,
istype-void,
istype-le,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermMultiply_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_functionality,
le_weakening,
mul_preserves_le,
le_wf,
squash_wf,
true_wf,
absval_mul,
subtype_rel_self,
iff_weakening_equal,
int-triangle-inequality,
int_subtype_base,
decidable__equal_int,
decidable__lt,
istype-less_than,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
le_transitivity,
add_functionality_wrt_le,
nat_plus_subtype_nat,
add_functionality_wrt_eq,
absval_pos,
absval-diff-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
thin,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
sqequalRule,
extract_by_obid,
isectElimination,
because_Cache,
functionEquality,
intEquality,
productEquality,
independent_isectElimination,
dependent_pairFormation_alt,
instantiate,
cumulativity,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
setElimination,
rename,
equalityIstype,
productElimination,
multiplyEquality,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination,
applyLambdaEquality,
functionExtensionality,
promote_hyp,
voidElimination,
universeIsType,
axiomEquality,
functionIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
isectIsType,
productIsType,
approximateComputation,
int_eqEquality,
independent_pairFormation,
universeEquality
Latex:
\mforall{}[f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbB{}]. \mforall{}[b:\mBbbR{}]. \mforall{}[a:\mBbbR{} supposing \mexists{}n:\mBbbN{}\msupplus{}. (\muparrow{}(f n))].
case-real3-seq(a;b;f) \mmember{} \{s:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}| 3-regular-seq(s)\} supposing \mforall{}n,m:\mBbbN{}\msupplus{}. ((\muparrow{}(f n)) {}\mRightarrow{} (\mneg{}\muparrow{}(f m)) \000C{}\mRightarrow{} (|(a m) - b m| \mleq{} 4))
Date html generated:
2019_10_29-AM-09_37_25
Last ObjectModification:
2019_06_14-PM-03_10_11
Theory : reals
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