Nuprl Lemma : qexp-difference-bound
∀[a,b:ℚ]. ∀n:ℕ+. (|a ↑ n - b ↑ n| ≤ (|a - b| * n * qmax(|a|;|b|) ↑ n - 1))
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qabs: |r|,
qmax: qmax(x;y),
qle: r ≤ s,
qsub: r - s,
qmul: r * s,
rationals: ℚ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
true: True,
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
subtract: n - m,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
qge: a ≥ b,
sq_type: SQType(T)
Lemmas referenced :
nat_plus_wf,
qle_witness,
qabs_wf,
qsub_wf,
qexp_wf,
nat_plus_subtype_nat,
qmul_wf,
subtype_rel_set,
rationals_wf,
less_than_wf,
int-subtype-rationals,
qmax_wf,
subtract_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_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_less_lemma,
int_formula_prop_wf,
le_wf,
qsum_wf,
int_seg_subtype_nat,
false_wf,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
int_seg_wf,
qmul_preserves_qle2,
qabs-nonneg,
qle_wf,
squash_wf,
true_wf,
qexp-difference-factor,
qabs-qmul,
iff_weakening_equal,
qabs-qsum-qle,
equal_wf,
qexp-add,
nat_wf,
add-commutes,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-mul-special,
zero-mul,
zero-add,
not-le-2,
less-iff-le,
condition-implies-le,
minus-minus,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
zero-qle-qabs,
qmax_ub,
qexp-nonneg,
qexp-qabs,
qexp_preserves_qle,
qle_reflexivity,
qle_functionality_wrt_implies,
qle_weakening_eq_qorder,
qmul_functionality_wrt_qle,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
applyEquality,
because_Cache,
intEquality,
natural_numberEquality,
independent_isectElimination,
dependent_set_memberEquality,
setElimination,
rename,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
addEquality,
productElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
hyp_replacement,
applyLambdaEquality,
minusEquality,
inlFormation,
inrFormation,
instantiate,
cumulativity
Latex:
\mforall{}[a,b:\mBbbQ{}]. \mforall{}n:\mBbbN{}\msupplus{}. (|a \muparrow{} n - b \muparrow{} n| \mleq{} (|a - b| * n * qmax(|a|;|b|) \muparrow{} n - 1))
Date html generated:
2018_05_22-AM-00_26_30
Last ObjectModification:
2017_07_26-PM-06_56_24
Theory : rationals
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