### 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

Home Index