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,
subtype_rel: A ⊆r B,
nat: ℕ,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
nat_plus: ℕ+,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
subtract: n - m,
squash: ↓T,
true: True,
guard: {T},
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),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
le_witness_for_triv,
istype-le,
absval_wf,
subtract_wf,
exp_wf2,
nat_plus_subtype_nat,
nat_plus_wf,
istype-nat,
istype-int,
decidable__le,
le_wf,
exp-convex,
nat_plus_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMinus_wf,
itermVar_wf,
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_minus_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformimplies_wf,
int_formual_prop_imp_lemma,
minus-one-mul,
squash_wf,
true_wf,
absval_sym,
subtype_rel_self,
iff_weakening_equal,
minus-minus,
minus-add,
eq_int_wf,
modulus_wf_int_mod,
istype-less_than,
subtype_rel_set,
int-subtype-int_mod,
eqtt_to_assert,
assert_of_eq_int,
int_mod_wf,
less_than_wf,
eqff_to_assert,
set_subtype_base,
int_subtype_base,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
neg_assert_of_eq_int,
equal_wf,
istype-universe,
mul-associates,
one-mul,
absval-minus,
exp-minus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
independent_isectElimination,
sqequalRule,
Error :productIsType,
hypothesisEquality,
applyEquality,
because_Cache,
Error :lambdaEquality_alt,
setElimination,
rename,
Error :inhabitedIsType,
Error :functionIsType,
natural_numberEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
Error :universeIsType,
dependent_functionElimination,
unionElimination,
dependent_set_memberEquality,
independent_functionElimination,
Error :dependent_set_memberEquality_alt,
minusEquality,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
hyp_replacement,
intEquality,
closedConclusion,
Error :lambdaFormation_alt,
equalityElimination,
Error :equalityIsType4,
baseApply,
promote_hyp,
cumulativity,
addEquality,
multiplyEquality,
Error :equalityIsType1
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:
2019_06_20-PM-02_31_09
Last ObjectModification:
2018_10_17-PM-00_25_14
Theory : num_thy_1
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