Nuprl Lemma : exp-convex
∀[a,b,c:ℕ]. ∀[n:ℕ+]. |a - b| ≤ c supposing |a^n - b^n| ≤ c^n
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,
subtract: n - m
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat_plus: ℕ+,
implies: P ⇒ Q,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
squash: ↓T,
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
less_than: a < b
Lemmas referenced :
nat_plus_properties,
less_than'_wf,
absval_wf,
subtract_wf,
le_wf,
exp_wf2,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
isect_wf,
primrec-wf-nat-plus,
nat_plus_subtype_nat,
nat_plus_wf,
nat_wf,
false_wf,
decidable__lt,
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,
add-subtract-cancel,
squash_wf,
true_wf,
exp1,
iff_weakening_equal,
exp_step,
multiply-is-int-iff,
absval-diff-product-bound,
exp_wf4,
exp_preserves_le,
absval-diff-symmetry,
subtype_base_sq,
int_subtype_base,
equal_wf,
imax_unfold,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
imin_unfold,
exp_preserves_lt,
set_subtype_base,
itermMultiply_wf,
int_term_value_mul_lemma,
mul_preserves_lt,
less_than_transitivity2,
mul_preserves_le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
voidElimination,
because_Cache,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
cumulativity,
universeEquality,
isectEquality,
independent_functionElimination,
minusEquality,
addLevel,
imageElimination,
imageMemberEquality,
baseClosed,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
multiplyEquality,
instantiate,
equalityElimination,
applyLambdaEquality
Latex:
\mforall{}[a,b,c:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. |a - b| \mleq{} c supposing |a\^{}n - b\^{}n| \mleq{} c\^{}n
Date html generated:
2018_05_21-PM-01_05_39
Last ObjectModification:
2018_01_28-PM-02_01_56
Theory : num_thy_1
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