Nuprl Lemma : exp_difference_bound
∀[n:ℕ+]. ∀[M:ℕ]. ∀[x,y:ℤ]. |x^n - y^n| ≤ ((n * M^n - 1) * |x - y|) supposing (|x| ≤ M) ∧ (|y| ≤ M)
Proof
Definitions occuring in Statement :
exp: i^n,
absval: |i|,
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
multiply: n * m,
subtract: n - m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
nat_plus: ℕ+,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
less_than': less_than'(a;b),
so_apply: x[s],
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
subtract: n - m
Lemmas referenced :
add-zero,
zero-mul,
add-mul-special,
add-commutes,
add-swap,
minus-one-mul,
add-associates,
minus-add,
exp_add,
multiply_functionality_wrt_le,
exp_wf4,
exp_preserves_le,
absval_exp,
le_weakening,
absval_sum,
le_functionality,
sum_le,
sum_constant,
mul_com,
iff_weakening_equal,
absval_mul,
exp_difference_factor,
true_wf,
squash_wf,
mul_preserves_le,
int_seg_wf,
false_wf,
int_seg_subtype_nat,
int_term_value_add_lemma,
itermAdd_wf,
int_seg_properties,
sum_wf,
nat_plus_wf,
nat_wf,
and_wf,
nat_plus_subtype_nat,
absval_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
nat_properties,
subtract_wf,
exp_wf2,
less_than'_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
lemma_by_obid,
isectElimination,
multiplyEquality,
dependent_set_memberEquality,
setElimination,
rename,
natural_numberEquality,
hypothesis,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
lambdaFormation,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination,
setEquality,
minusEquality
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[M:\mBbbN{}]. \mforall{}[x,y:\mBbbZ{}]. |x\^{}n - y\^{}n| \mleq{} ((n * M\^{}n - 1) * |x - y|) supposing (|x| \mleq{} M) \mwedge{} (|y| \mleq{} M)
Date html generated:
2016_05_14-PM-04_28_04
Last ObjectModification:
2016_01_14-PM-11_40_37
Theory : num_thy_1
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