### Nuprl Lemma : int-sq-root

`∀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` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` and: `P ∧ Q` uall: `∀[x:A]. B[x]` prop: `ℙ` guard: `{T}` so_lambda: `λ2x.t[x]` nat: `ℕ` so_apply: `x[s]` implies: `P `` Q` nat_plus: `ℕ+` nequal: `a ≠ b ∈ T ` not: `¬A` uimplies: `b supposing a` sq_type: `SQType(T)` false: `False` sq_exists: `∃x:A [B[x]]` le: `A ≤ B` cand: `A c∧ B` int_nzero: `ℤ-o` subtype_rel: `A ⊆r B` decidable: `Dec(P)` or: `P ∨ Q` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top`
Lemmas referenced :  div_nat_induction-ext less_than_wf sq_exists_wf nat_wf le_wf subtype_base_sq int_subtype_base equal-wf-base true_wf nat_plus_wf false_wf div_rem_sum nequal_wf rem_bounds_1 nat_plus_subtype_nat equal_wf set-value-type int-value-type decidable__lt nat_properties nat_plus_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformeq_wf itermMultiply_wf intformless_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_term_value_mul_lemma int_formula_prop_less_lemma int_term_value_add_lemma int_formula_prop_wf
Rules used in proof :  cut introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity dependent_functionElimination thin dependent_set_memberEquality natural_numberEquality sqequalRule independent_pairFormation imageMemberEquality hypothesisEquality baseClosed hypothesis isectElimination lambdaEquality productEquality multiplyEquality setElimination rename because_Cache addEquality independent_functionElimination lambdaFormation divideEquality addLevel instantiate cumulativity intEquality independent_isectElimination equalityTransitivity equalitySymmetry voidElimination dependent_set_memberFormation productElimination applyEquality cutEval unionElimination imageElimination approximateComputation dependent_pairFormation int_eqEquality isect_memberEquality voidEquality promote_hyp

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_44
Last ObjectModification: 2017_11_20-PM-00_43_58

Theory : general

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