Nuprl Lemma : assert-regular-upto
∀[k,n:ℕ]. ∀[f:ℕ+ ⟶ ℤ]. (↑regular-upto(k;n;f) ⇐⇒ ∀i,j:ℕ+n + 1. (|(i * (f j)) - j * (f i)| ≤ ((2 * k) * (i + j))))
Proof
Definitions occuring in Statement :
regular-upto: regular-upto(k;n;f),
absval: |i|,
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
assert: ↑b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
nat: ℕ,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
nat_plus: ℕ+,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
so_apply: x[s],
regular-upto: regular-upto(k;n;f),
subtract: n - m,
top: Top,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
sq_type: SQType(T),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
int_seg_wf,
assert_wf,
regular-upto_wf,
nat_plus_wf,
all_wf,
le_wf,
absval_wf,
subtract_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
less_than'_wf,
assert_witness,
nat_wf,
assert-bdd-all,
bdd-all_wf,
le_int_wf,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-associates,
add-zero,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
lelt_wf,
assert_of_le_int,
minus-minus,
add-swap,
multiply-is-int-iff,
set_subtype_base,
int_subtype_base,
add-is-int-iff,
subtype_base_sq,
int_seg_subtype_nat_plus,
and_wf,
equal_wf,
add-member-int_seg2,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
functionExtensionality,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
multiplyEquality,
dependent_set_memberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
voidElimination,
independent_functionElimination,
independent_isectElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
intEquality,
isect_memberEquality,
addLevel,
voidEquality,
minusEquality,
allFunctionality,
levelHypothesis,
promote_hyp,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
hyp_replacement,
baseApply,
closedConclusion,
baseClosed,
instantiate,
cumulativity,
applyLambdaEquality,
allLevelFunctionality
Latex:
\mforall{}[k,n:\mBbbN{}]. \mforall{}[f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}].
(\muparrow{}regular-upto(k;n;f) \mLeftarrow{}{}\mRightarrow{} \mforall{}i,j:\mBbbN{}\msupplus{}n + 1. (|(i * (f j)) - j * (f i)| \mleq{} ((2 * k) * (i + j))))
Date html generated:
2017_10_03-AM-08_42_39
Last ObjectModification:
2017_09_06-PM-04_04_24
Theory : reals
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