Nuprl Lemma : integer-nth-root
∀n:ℕ+. ∀x:ℕ. (∃r:ℕ [((r^n ≤ x) ∧ x < (r + 1)^n)])
Proof
Definitions occuring in Statement :
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
le: A ≤ B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uiff: uiff(P;Q),
sq_type: SQType(T),
sq_stable: SqStable(P),
le: A ≤ B,
less_than': less_than'(a;b),
cand: A c∧ B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
subtype_rel: A ⊆r B,
sq_exists: ∃x:A [B[x]],
ge: i ≥ j ,
guard: {T},
squash: ↓T,
int_upper: {i...},
and: P ∧ Q,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
nat_plus: ℕ+,
nat: ℕ,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
subtype_rel_sets,
set_subtype_base,
mul_preserves_le,
false_wf,
multiply-is-int-iff,
add-is-int-iff,
decidable__equal_int,
subtype_base_sq,
int_term_value_mul_lemma,
int_term_value_add_lemma,
itermMultiply_wf,
itermAdd_wf,
sq_stable__less_than,
exp-fastexp,
fastexp_wf,
decidable__lt,
rem_bounds_1,
div_rem_sum,
nat_plus_subtype_nat,
exp-of-mul,
istype-false,
squash_wf,
true_wf,
exp-zero,
subtype_rel_self,
iff_weakening_equal,
exp-positive,
nat_plus_wf,
int_subtype_base,
int_formula_prop_eq_lemma,
intformeq_wf,
nequal_wf,
subtype_rel_sets_simple,
divide_wfa,
istype-nat,
nat_properties,
le_wf,
nat_wf,
sq_exists_wf,
div_nat_induction,
int-value-type,
equal_wf,
set-value-type,
istype-less_than,
exp-ge-1,
istype-le,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_plus_properties,
exp_wf2,
istype-int,
less_than_wf,
set_wf
Rules used in proof :
divideEquality,
Error :equalityIsType4,
setEquality,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
cumulativity,
multiplyEquality,
Error :dependent_set_memberFormation_alt,
instantiate,
universeEquality,
productElimination,
sqequalBase,
applyEquality,
Error :productIsType,
Error :setIsType,
addEquality,
because_Cache,
imageElimination,
baseClosed,
imageMemberEquality,
applyLambdaEquality,
productEquality,
Error :inhabitedIsType,
Error :equalityIstype,
equalitySymmetry,
equalityTransitivity,
cutEval,
Error :universeIsType,
independent_pairFormation,
voidElimination,
Error :isect_memberEquality_alt,
int_eqEquality,
Error :dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
rename,
setElimination,
Error :dependent_set_memberEquality_alt,
hypothesis,
hypothesisEquality,
natural_numberEquality,
Error :lambdaEquality_alt,
sqequalRule,
intEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}x:\mBbbN{}. (\mexists{}r:\mBbbN{} [((r\^{}n \mleq{} x) \mwedge{} x < (r + 1)\^{}n)])
Date html generated:
2019_06_20-PM-02_33_36
Last ObjectModification:
2019_06_19-AM-11_28_46
Theory : num_thy_1
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