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 :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
int_upper: {i...},
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
nat: ℕ,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
squash: ↓T,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
int_nzero: ℤ-o,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
sq_type: SQType(T)
Lemmas referenced :
set_wf,
less_than_wf,
exp_wf2,
nat_plus_subtype_nat,
exp-ge-1,
false_wf,
le_wf,
equal_wf,
set-value-type,
int-value-type,
div_nat_induction,
sq_exists_wf,
nat_wf,
nat_properties,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
equal-wf-T-base,
set_subtype_base,
nat_plus_wf,
squash_wf,
true_wf,
exp-zero,
iff_weakening_equal,
exp-positive,
exp-of-mul,
div_rem_sum,
subtype_rel_sets,
nequal_wf,
rem_bounds_1,
decidable__lt,
not-lt-2,
less-iff-le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-swap,
add-commutes,
zero-add,
le-add-cancel,
fastexp_wf,
exp-fastexp,
sq_stable__less_than,
decidable__le,
intformnot_wf,
intformle_wf,
itermAdd_wf,
itermMultiply_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
subtype_base_sq,
decidable__equal_int,
add-is-int-iff,
multiply-is-int-iff,
mul_preserves_le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
sqequalRule,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
applyEquality,
independent_pairFormation,
cutEval,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
setElimination,
rename,
dependent_functionElimination,
productEquality,
because_Cache,
addEquality,
independent_functionElimination,
divideEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
productElimination,
dependent_set_memberFormation,
universeEquality,
multiplyEquality,
setEquality,
unionElimination,
instantiate,
cumulativity,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}x:\mBbbN{}. (\mexists{}r:\mBbbN{} [((r\^{}n \mleq{} x) \mwedge{} x < (r + 1)\^{}n)])
Date html generated:
2018_05_21-PM-07_50_08
Last ObjectModification:
2017_07_26-PM-05_27_54
Theory : general
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