### Nuprl Lemma : isqrt_newton_wf

`∀n,x:ℕ+.  isqrt_newton(n;x) ∈ ∃r:ℕ [(((r * r) ≤ n) ∧ n < (r + 1) * (r + 1))] supposing n < (x + 1) * (x + 1)`

Proof

Definitions occuring in Statement :  isqrt_newton: `isqrt_newton(n;x)` nat_plus: `ℕ+` nat: `ℕ` less_than: `a < b` uimplies: `b supposing a` le: `A ≤ B` all: `∀x:A. B[x]` sq_exists: `∃x:A [B[x]]` and: `P ∧ Q` member: `t ∈ T` multiply: `n * m` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` nat: `ℕ` uall: `∀[x:A]. B[x]` nat_plus: `ℕ+` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` ge: `i ≥ j ` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` isqrt_newton: `isqrt_newton(n;x)` nequal: `a ≠ b ∈ T ` has-value: `(a)↓` less_than: `a < b` int_nzero: `ℤ-o` so_lambda: `λ2x.t[x]` so_apply: `x[s]` true: `True` sq_type: `SQType(T)` squash: `↓T` subtract: `n - m` sq_exists: `∃x:A [B[x]]` cand: `A c∧ B` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` bfalse: `ff` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  subtract_wf nat_plus_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermMultiply_wf itermAdd_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_mul_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf le_wf less_than_wf nat_plus_wf nat_properties ge_wf int_seg_wf int_seg_properties decidable__equal_int int_seg_subtype false_wf intformeq_wf int_formula_prop_eq_lemma equal-wf-base int_subtype_base value-type-has-value int-value-type mul_preserves_eq equal_wf decidable__lt lelt_wf nat_wf div_rem_sum2 subtype_rel_sets nequal_wf rem_bounds_1 nat_plus_subtype_nat div_bounds_1 subtype_base_sq true_wf mul-distributes mul-commutes add-commutes mul_preserves_le minus-one-mul mul-swap mul_cancel_in_lt eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int lt_int_wf assert_of_lt_int top_wf square_non_neg multiply-is-int-iff add-is-int-iff mul-distributes-right add-associates mul-associates one-mul two-mul less_than_functionality le_weakening multiply_functionality_wrt_le mul_preserves_lt
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation introduction hypothesis sqequalHypSubstitution dependent_functionElimination thin dependent_set_memberEquality extract_by_obid isectElimination multiplyEquality addEquality setElimination rename because_Cache natural_numberEquality hypothesisEquality unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination equalityTransitivity equalitySymmetry axiomEquality intWeakElimination productElimination applyEquality applyLambdaEquality hypothesis_subsumption divideEquality baseClosed callbyvalueReduce setEquality addLevel instantiate cumulativity imageElimination imageMemberEquality productEquality lessEquality equalityElimination int_eqReduceTrueSq promote_hyp int_eqReduceFalseSq lessCases sqequalAxiom pointwiseFunctionality baseApply closedConclusion minusEquality

Latex:
\mforall{}n,x:\mBbbN{}\msupplus{}.
isqrt\_newton(n;x)  \mmember{}  \mexists{}r:\mBbbN{}  [(((r  *  r)  \mleq{}  n)  \mwedge{}  n  <  (r  +  1)  *  (r  +  1))]  supposing  n  <  (x  +  1)  *  (x  +  1)

Date html generated: 2018_05_21-PM-07_51_49
Last ObjectModification: 2017_07_26-PM-05_29_27

Theory : general

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