Nuprl Lemma : integer-nth-root-ext
∀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 :
member: t ∈ T,
so_apply: x[s1;s2],
natrec: natrec,
genrec: genrec,
genrec-ap: genrec-ap,
integer-nth-root,
div_nat_induction,
rem_bounds_1,
decidable__lt,
decidable__equal_int,
decidable__squash,
decidable__and,
decidable__less_than',
decidable__int_equal,
decidable_functionality,
squash_elim,
sq_stable_from_decidable,
any: any x,
iff_preserves_decidability,
sq_stable__from_stable,
stable__from_decidable,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
uimplies: b supposing a
Lemmas referenced :
integer-nth-root,
lifting-strict-int_eq,
istype-void,
strict4-decide,
lifting-strict-decide,
lifting-strict-less,
div_nat_induction,
rem_bounds_1,
decidable__lt,
decidable__equal_int,
decidable__squash,
decidable__and,
decidable__less_than',
decidable__int_equal,
decidable_functionality,
squash_elim,
sq_stable_from_decidable,
iff_preserves_decidability,
sq_stable__from_stable,
stable__from_decidable
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
Error :isect_memberEquality_alt,
voidElimination,
independent_isectElimination
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_42
Last ObjectModification:
2019_04_15-PM-10_31_53
Theory : num_thy_1
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