Nuprl Lemma : int-sq-root

x:ℕ(∃r:ℕ [(((r r) ≤ x) ∧ x < (r 1) (r 1))])


Proof




Definitions occuring in Statement :  nat: less_than: a < b le: A ≤ B all: x:A. B[x] sq_exists: x:A [B[x]] and: P ∧ Q multiply: m add: 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: guard: {T} so_lambda: λ2x.t[x] nat: so_apply: x[s] implies:  Q nat_plus: + nequal: a ≠ b ∈  not: ¬A uimplies: supposing a sq_type: SQType(T) false: False sq_exists: x:A [B[x]] le: A ≤ B cand: c∧ B int_nzero: -o subtype_rel: A ⊆B decidable: Dec(P) or: P ∨ Q ge: i ≥  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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