### Nuprl Lemma : integer-sqrt-newton

`∀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 :  all: `∀x:A. B[x]` member: `t ∈ T` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` uall: `∀[x:A]. B[x]` uimplies: `b supposing a` sq_type: `SQType(T)` implies: `P `` Q` guard: `{T}` and: `P ∧ Q` cand: `A c∧ B` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` less_than: `a < b` squash: `↓T` true: `True` sq_exists: `∃x:{A| B[x]}` nat_plus: `ℕ+` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtype_rel: `A ⊆r B` top: `Top` subtract: `n - m` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut lemma_by_obid sqequalHypSubstitution dependent_functionElimination thin setElimination rename hypothesisEquality natural_numberEquality hypothesis unionElimination instantiate isectElimination cumulativity intEquality independent_isectElimination because_Cache independent_functionElimination introduction sqequalRule independent_pairFormation multiplyEquality equalityTransitivity equalitySymmetry addEquality imageMemberEquality baseClosed dependent_set_memberEquality productElimination voidElimination applyEquality lambdaEquality isect_memberEquality voidEquality minusEquality dependent_pairFormation int_eqEquality computeAll

Latex:
\mforall{}x:\mBbbN{}.  (\mexists{}r:\{\mBbbN{}|  (((r  *  r)  \mleq{}  x)  \mwedge{}  x  <  (r  +  1)  *  (r  +  1))\})

Date html generated: 2016_05_15-PM-05_16_42
Last ObjectModification: 2016_01_16-AM-11_38_14

Theory : general

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