Nuprl Lemma : exp-difference-inequality
∀[n:ℕ+]. ∀[a,b:ℕ]. (((a + b)^n - a^n) ≤ (n * b * (a + b)^(n - 1)))
Proof
Definitions occuring in Statement :
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
le: A ≤ B,
and: P ∧ Q,
uimplies: b supposing a,
nat: ℕ,
true: True,
nat_plus: ℕ+,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
squash: ↓T,
less_than': less_than'(a;b),
guard: {T},
so_lambda: λ2x.t[x],
int_seg: {i..j-},
int_iseg: {i...j},
so_apply: x[s],
lelt: i ≤ j < k,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
subtract: n - m,
uiff: uiff(P;Q),
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
bor: p ∨bq,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
ycomb: Y,
choose: choose(n;i)
Lemmas referenced :
binomial-int,
nat_plus_subtype_nat,
le_witness_for_triv,
nat_wf,
nat_plus_wf,
exp_wf2,
subtract_wf,
nat_properties,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
sum_wf,
add_nat_wf,
istype-false,
le_weakening2,
subtract-add-cancel,
decidable__lt,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
choose_wf,
subtype_rel_sets,
lelt_wf,
istype-less_than,
int_seg_subtype_nat,
subtract_nat_wf,
int_seg_properties,
int_seg_wf,
iff_weakening_equal,
sum_scalar_mult,
squash_wf,
true_wf,
subtype_rel_self,
subtype_base_sq,
int_subtype_base,
satisfiable-full-omega-tt,
less_than_wf,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
less-iff-le,
not-lt-2,
false_wf,
equal_wf,
sum_split1,
int_term_value_mul_lemma,
itermMultiply_wf,
set_subtype_base,
exp0_lemma,
decidable__equal_int,
add-subtract-cancel,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
mul-swap,
add-swap,
exp_step,
sum_le,
choose-inequality1,
exp_wf4,
mul_bounds_1a,
mul_preserves_le,
multiply-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
universeIsType,
setElimination,
rename,
natural_numberEquality,
multiplyEquality,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
imageElimination,
addEquality,
lambdaFormation_alt,
applyLambdaEquality,
equalityIsType1,
intEquality,
closedConclusion,
productEquality,
setIsType,
productIsType,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
cumulativity,
computeAll,
dependent_pairFormation,
setEquality,
minusEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
lambdaFormation,
dependent_set_memberEquality,
promote_hyp,
equalityElimination,
functionIsType,
baseApply,
pointwiseFunctionality,
hyp_replacement
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[a,b:\mBbbN{}]. (((a + b)\^{}n - a\^{}n) \mleq{} (n * b * (a + b)\^{}(n - 1)))
Date html generated:
2019_10_15-AM-11_23_39
Last ObjectModification:
2018_10_16-PM-03_16_36
Theory : general
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