Nuprl Lemma : binomial-inequality1
∀[a,b:ℕ]. ∀[n:ℕ+]. ((a^n + b^n) ≤ (a + b)^n)
Proof
Definitions occuring in Statement :
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
add: n + m
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
squash: ↓T,
nat_plus: ℕ+,
le: A ≤ B,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
top: Top,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
int_iseg: {i...j},
so_apply: x[s],
lelt: i ≤ j < k,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
choose: choose(n;i),
ycomb: Y,
eq_int: (i =z j),
btrue: tt,
bor: p ∨bq,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
binomial-int,
nat_plus_subtype_nat,
le_wf,
exp_wf2,
nat_plus_wf,
nat_wf,
squash_wf,
true_wf,
sum_split1,
decidable__lt,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
choose_wf,
subtype_rel_sets,
lelt_wf,
int_seg_properties,
nat_plus_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
int_seg_subtype_nat,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
int_seg_wf,
iff_weakening_equal,
sum_wf,
add-subtract-cancel,
add-swap,
add_functionality_wrt_eq,
sum_split_first,
non_neg_sum,
mul_bounds_1a,
multiply_nat_wf,
exp_wf4,
le_weakening2,
le_reflexive,
and_wf,
equal_wf,
minus-minus,
add-mul-special,
zero-mul,
le_functionality,
le_weakening,
add-is-int-iff,
set_subtype_base,
int_subtype_base,
minus-zero,
exp0_lemma,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
mul-associates,
mul-commutes,
one-mul
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
applyEquality,
sqequalRule,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
addEquality,
because_Cache,
lambdaEquality,
imageElimination,
equalityTransitivity,
intEquality,
dependent_set_memberEquality,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isect_memberEquality,
voidEquality,
minusEquality,
multiplyEquality,
productEquality,
setEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
imageMemberEquality,
baseClosed,
universeEquality,
baseApply,
closedConclusion,
equalityElimination,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[a,b:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. ((a\^{}n + b\^{}n) \mleq{} (a + b)\^{}n)
Date html generated:
2018_05_21-PM-08_27_50
Last ObjectModification:
2017_07_26-PM-05_55_23
Theory : general
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