Nuprl Lemma : int-sqrt-ext
∀x:ℕ. (∃r:ℕ [(((r * r) ≤ x) ∧ x < (r + 1) * (r + 1))])
Proof
Definitions occuring in Statement :
nat: ℕ,
less_than: a < b,
le: A ≤ B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
genrec-ap: genrec-ap,
decidable__less_than',
decidable__and,
iff_preserves_decidability,
decidable_functionality,
so_apply: x[s],
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
squash: ↓T,
or: P ∨ Q,
guard: {T},
prop: ℙ,
has-value: (a)↓,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
strict4: strict4(F),
uimplies: b supposing a,
top: Top,
so_apply: x[s1;s2;s3;s4],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
uall: ∀[x:A]. B[x],
decidable__squash,
decidable__lt,
genrec: genrec,
natrec: natrec,
div_nat_induction-ext,
int-sq-root,
member: t ∈ T
Lemmas referenced :
lifting-strict-less,
lifting-strict-decide,
equal_wf,
top_wf,
is-exception_wf,
base_wf,
has-value_wf_base,
lifting-strict-spread,
int-sq-root,
decidable__less_than',
decidable__and,
iff_preserves_decidability,
decidable_functionality,
decidable__squash,
decidable__lt,
div_nat_induction-ext
Rules used in proof :
because_Cache,
decideExceptionCases,
independent_functionElimination,
dependent_functionElimination,
sqleReflexivity,
unionElimination,
unionEquality,
equalitySymmetry,
equalityTransitivity,
callbyvalueDecide,
inlFormation,
exceptionSqequal,
imageElimination,
imageMemberEquality,
inrFormation,
applyExceptionCases,
hypothesisEquality,
closedConclusion,
baseApply,
callbyvalueApply,
lambdaFormation,
independent_pairFormation,
independent_isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
baseClosed,
isectElimination,
sqequalHypSubstitution,
thin,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction
Latex:
\mforall{}x:\mBbbN{}. (\mexists{}r:\mBbbN{} [(((r * r) \mleq{} x) \mwedge{} x < (r + 1) * (r + 1))])
Date html generated:
2018_05_21-PM-07_49_54
Last ObjectModification:
2018_05_19-AM-07_44_38
Theory : general
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