Nuprl Lemma : exp-convex2
∀[a,b:ℤ]. ∀[c:ℕ]. ∀[n:ℕ+]. |a - b| ≤ c supposing (|a^n - b^n| ≤ c^n) ∧ (0 ≤ a ⇐⇒ 0 ≤ b)
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,
iff: P ⇐⇒ Q,
and: P ∧ Q,
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,
iff: P ⇐⇒ Q,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
squash: ↓T,
true: True,
less_than: a < b,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
subtract: n - m
Lemmas referenced :
less_than'_wf,
absval_wf,
subtract_wf,
le_wf,
exp_wf2,
nat_plus_subtype_nat,
iff_wf,
nat_plus_wf,
nat_wf,
decidable__le,
exp-convex,
nat_plus_properties,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMinus_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformimplies_wf,
int_formual_prop_imp_lemma,
squash_wf,
true_wf,
eq_int_wf,
modulus_wf_int_mod,
less_than_wf,
subtype_rel_set,
int_mod_wf,
int-subtype-int_mod,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
exp-minus,
iff_weakening_equal,
absval_sym,
minus-minus,
minus-add,
minus-one-mul,
minus-one-mul-top
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
natural_numberEquality,
isect_memberEquality,
intEquality,
voidElimination,
unionElimination,
independent_functionElimination,
dependent_set_memberEquality,
independent_isectElimination,
minusEquality,
dependent_pairFormation,
int_eqEquality,
voidEquality,
independent_pairFormation,
computeAll,
hyp_replacement,
imageElimination,
imageMemberEquality,
baseClosed,
lambdaFormation,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
equalityEquality,
addEquality
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[c:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. |a - b| \mleq{} c supposing (|a\^{}n - b\^{}n| \mleq{} c\^{}n) \mwedge{} (0 \mleq{} a \mLeftarrow{}{}\mRightarrow{} 0 \mleq{} b)
Date html generated:
2016_10_25-AM-10_59_36
Last ObjectModification:
2016_07_12-AM-07_06_49
Theory : general
Home
Index