Nuprl Lemma : general-iroot-property
∀[n:ℕ+]. ∀[x:ℤ].
(((0 ≤ x) ⇒ ((general-iroot(n;x)^n ≤ x) ∧ x < (general-iroot(n;x) + 1)^n))
∧ ((x < 0 ∧ ((n mod 2) = 1 ∈ ℤ)) ⇒ ((x ≤ general-iroot(n;x)^n) ∧ (general-iroot(n;x) - 1)^n < x))
∧ ((x < 0 ∧ ((n mod 2) = 0 ∈ ℤ)) ⇒ (general-iroot(n;x) = 0 ∈ ℤ)))
Proof
Definitions occuring in Statement :
general-iroot: general-iroot(n;x),
exp: i^n,
modulus: a mod n,
nat_plus: ℕ+,
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
implies: P ⇒ Q,
and: P ∧ Q,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
general-iroot: general-iroot(n;x),
prop: ℙ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
has-value: (a)↓,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
le: A ≤ B,
cand: A c∧ B,
nat: ℕ,
int_lower: {...i},
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
sq_stable: SqStable(P)
Lemmas referenced :
le_wf,
less_than_wf,
equal-wf-T-base,
modulus_wf_int_mod,
subtype_rel_set,
int_mod_wf,
int-subtype-int_mod,
value-type-has-value,
nat_plus_wf,
set-value-type,
int-value-type,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
eq_int_wf,
assert_of_eq_int,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
member-less_than,
less_than'_wf,
exp_wf2,
nat_plus_subtype_nat,
general-iroot_wf,
subtract_wf,
intformless_wf,
intformle_wf,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
iroot-property,
int_subtype_base,
integer-nth-root2,
all_wf,
int_lower_wf,
sq_exists_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
squash_wf,
sq_stable__and,
sq_stable__le,
sq_stable__less_than
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
hypothesis,
productEquality,
because_Cache,
applyEquality,
sqequalRule,
intEquality,
lambdaEquality,
independent_isectElimination,
baseClosed,
callbyvalueReduce,
productElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
lessCases,
axiomSqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
imageMemberEquality,
imageElimination,
independent_functionElimination,
dependent_set_memberEquality,
int_eqReduceTrueSq,
setElimination,
rename,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
computeAll,
promote_hyp,
instantiate,
cumulativity,
int_eqReduceFalseSq,
independent_pairEquality,
axiomEquality,
addEquality,
setEquality
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[x:\mBbbZ{}].
(((0 \mleq{} x) {}\mRightarrow{} ((general-iroot(n;x)\^{}n \mleq{} x) \mwedge{} x < (general-iroot(n;x) + 1)\^{}n))
\mwedge{} ((x < 0 \mwedge{} ((n mod 2) = 1)) {}\mRightarrow{} ((x \mleq{} general-iroot(n;x)\^{}n) \mwedge{} (general-iroot(n;x) - 1)\^{}n < x))
\mwedge{} ((x < 0 \mwedge{} ((n mod 2) = 0)) {}\mRightarrow{} (general-iroot(n;x) = 0)))
Date html generated:
2019_06_20-PM-02_35_08
Last ObjectModification:
2019_03_19-AM-10_49_49
Theory : num_thy_1
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