Nuprl Lemma : integer-nth-root2
∀n:{n:ℕ+| (n mod 2) = 1 ∈ ℤ} . ∀x:{...0}. (∃r:{...0} [((r - 1)^n < x ∧ (x ≤ r^n))])
Proof
Definitions occuring in Statement :
exp: i^n,
modulus: a mod n,
int_lower: {...i},
nat_plus: ℕ+,
less_than: a < b,
le: A ≤ B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
set: {x:A| B[x]} ,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
int_lower: {...i},
uall: ∀[x:A]. B[x],
guard: {T},
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
less_than: a < b,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
ge: i ≥ j ,
cand: A c∧ B,
sq_type: SQType(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
bfalse: ff,
le: A ≤ B
Lemmas referenced :
subtract_wf,
int_formula_prop_less_lemma,
intformless_wf,
exp_wf2,
decidable__lt,
set_subtype_base,
iff_weakening_equal,
nat_plus_subtype_nat,
nat_wf,
exp-minus,
true_wf,
squash_wf,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
itermSubtract_wf,
intformeq_wf,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
nat_properties,
int-subtype-int_mod,
int_mod_wf,
subtype_rel_set,
less_than_wf,
modulus_wf_int_mod,
equal-wf-T-base,
nat_plus_wf,
set_wf,
int_lower_wf,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_minus_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermMinus_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
sq_stable__equal,
int_lower_properties,
integer-nth-root-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
dependent_set_memberEquality,
minusEquality,
isectElimination,
natural_numberEquality,
hypothesis,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
introduction,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
comment,
applyEquality,
dependent_set_memberFormation,
productElimination,
instantiate,
addEquality,
universeEquality,
productEquality
Latex:
\mforall{}n:\{n:\mBbbN{}\msupplus{}| (n mod 2) = 1\} . \mforall{}x:\{...0\}. (\mexists{}r:\{...0\} [((r - 1)\^{}n < x \mwedge{} (x \mleq{} r\^{}n))])
Date html generated:
2019_06_20-PM-02_33_47
Last ObjectModification:
2019_03_19-AM-10_49_03
Theory : num_thy_1
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