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 le: A ≤ B sq_exists: x:{A| B[x]} 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 :  int_formula_prop_wf int_term_value_add_lemma int_formula_prop_less_lemma int_term_value_mul_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermAdd_wf intformless_wf itermMultiply_wf intformeq_wf itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_plus_properties nat_properties decidable__lt int-value-type set-value-type nat_plus_subtype_nat rem_bounds_1 nequal_wf div_rem_sum false_wf nat_plus_wf true_wf equal_wf int_subtype_base subtype_base_sq le_wf and_wf nat_wf sq_exists_wf less_than_wf div_nat_induction-ext
Rules used in proof :  cut lemma_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity dependent_functionElimination thin dependent_set_memberEquality natural_numberEquality sqequalRule independent_pairFormation introduction imageMemberEquality hypothesisEquality baseClosed hypothesis isectElimination lambdaEquality multiplyEquality setElimination rename addEquality independent_functionElimination lambdaFormation productEquality because_Cache divideEquality addLevel instantiate cumulativity intEquality independent_isectElimination equalityTransitivity equalitySymmetry voidElimination productElimination dependent_set_memberFormation applyEquality cutEval equalityEquality unionElimination imageElimination dependent_pairFormation int_eqEquality isect_memberEquality voidEquality computeAll promote_hyp

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_12_59
Last ObjectModification: 2016_01_16-AM-11_35_44

Theory : general


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