Nuprl Lemma : iroot-property
∀[n:ℕ+]. ∀[x:ℕ]. ((iroot(n;x)^n ≤ x) ∧ x < (iroot(n;x) + 1)^n)
Proof
Definitions occuring in Statement :
iroot: iroot(n;x),
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iroot: iroot(n;x),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
nat: ℕ,
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
implies: P ⇒ Q,
le: A ≤ B,
not: ¬A,
false: False,
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
integer-nth-root-ext,
subtype_rel_self,
nat_plus_wf,
all_wf,
nat_wf,
sq_exists_wf,
le_wf,
exp_wf2,
less_than_wf,
equal_wf,
member-less_than,
less_than'_wf,
nat_plus_subtype_nat,
iroot_wf,
squash_wf,
sq_stable__and,
sq_stable__le,
sq_stable__less_than
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
isectElimination,
functionEquality,
lambdaEquality,
productEquality,
hypothesisEquality,
because_Cache,
setElimination,
rename,
addEquality,
natural_numberEquality,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[x:\mBbbN{}]. ((iroot(n;x)\^{}n \mleq{} x) \mwedge{} x < (iroot(n;x) + 1)\^{}n)
Date html generated:
2019_06_20-PM-02_34_03
Last ObjectModification:
2019_03_19-AM-10_49_16
Theory : num_thy_1
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