### Nuprl Lemma : reg-seq-mul_wf2

`∀[x,y:ℝ].  (reg-seq-mul(x;y) ∈ {f:ℕ+ ⟶ ℤ| imax(|x 1|;|y 1|) + 4-regular-seq(f)} )`

Proof

Definitions occuring in Statement :  reg-seq-mul: `reg-seq-mul(x;y)` real: `ℝ` regular-int-seq: `k-regular-seq(f)` imax: `imax(a;b)` absval: `|i|` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` real: `ℝ` nat_plus: `ℕ+` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` prop: `ℙ` false: `False` subtype_rel: `A ⊆r B` nat: `ℕ` canon-bnd: `canon-bnd(x)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` le: `A ≤ B` int_upper: `{i...}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` cand: `A c∧ B` less_than': `less_than'(a;b)` true: `True` rev_uimplies: `rev_uimplies(P;Q)` ge: `i ≥ j ` sq_stable: `SqStable(P)` squash: `↓T`
Lemmas referenced :  reg-seq-mul_wf regular-int-seq_wf imax_wf absval_wf decidable__lt full-omega-unsat intformnot_wf intformless_wf itermConstant_wf istype-int int_formula_prop_not_lemma istype-void int_formula_prop_less_lemma int_term_value_constant_lemma int_formula_prop_wf istype-less_than real_wf ifthenelse_wf le_int_wf eqtt_to_assert assert_of_le_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf le_wf istype-le intformand_wf intformle_wf itermVar_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_term_value_add_lemma int_subtype_base decidable__equal_int intformeq_wf int_formula_prop_eq_lemma add-is-int-iff false_wf add_functionality_wrt_eq imax_unfold iff_weakening_equal reg-seq-mul-regular canon-bnd_wf imax_nat_plus subtype_rel_set int_upper_wf nat_plus_wf istype-int_upper subtype_rel_sets_simple less_than_wf istype-false not-lt-2 add_functionality_wrt_le add-commutes zero-add le-add-cancel nat_plus_properties imax_ub decidable__le mul_preserves_le nat_plus_subtype_nat le_functionality le_weakening sq_stable__le
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut dependent_set_memberEquality_alt extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename because_Cache hypothesis universeIsType addEquality applyEquality natural_numberEquality dependent_functionElimination unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt isect_memberEquality_alt voidElimination sqequalRule hypothesisEquality inhabitedIsType equalityTransitivity equalitySymmetry axiomEquality isectIsTypeImplies intEquality lambdaFormation_alt equalityElimination productElimination equalityIstype promote_hyp instantiate cumulativity int_eqEquality independent_pairFormation pointwiseFunctionality baseApply closedConclusion baseClosed sqequalIntensionalEquality functionEquality multiplyEquality applyLambdaEquality inlFormation_alt imageMemberEquality imageElimination inrFormation_alt

Latex:
\mforall{}[x,y:\mBbbR{}].    (reg-seq-mul(x;y)  \mmember{}  \{f:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}|  imax(|x  1|;|y  1|)  +  4-regular-seq(f)\}  )

Date html generated: 2019_10_16-PM-03_06_48
Last ObjectModification: 2019_01_31-PM-04_48_18

Theory : reals

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