Nuprl Lemma : sum_arith1

`∀[n:ℕ]. ∀[a,b:ℤ].  ((Σ(a + (b * i) | i < n) * 2) = (n * (a + a + (b * (n - 1)))) ∈ ℤ)`

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` nat: `ℕ` uall: `∀[x:A]. B[x]` multiply: `n * m` subtract: `n - m` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` nat: `ℕ` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtract: `n - m` decidable: `Dec(P)` or: `P ∨ Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` subtype_rel: `A ⊆r B` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  sum-as-primrec int_seg_wf nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf primrec0_lemma decidable__equal_int intformnot_wf intformeq_wf itermMultiply_wf itermAdd_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_mul_lemma int_term_value_add_lemma decidable__le subtract_wf itermSubtract_wf int_term_value_subtract_lemma primrec-unroll eq_int_wf bool_wf uiff_transitivity equal-wf-base int_subtype_base assert_wf eqtt_to_assert assert_of_eq_int iff_transitivity bnot_wf not_wf iff_weakening_uiff eqff_to_assert assert_of_bnot multiply-is-int-iff false_wf equal_wf nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality lambdaEquality addEquality multiplyEquality setElimination rename hypothesis natural_numberEquality lambdaFormation intWeakElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality because_Cache unionElimination minusEquality equalityElimination baseApply closedConclusion baseClosed applyEquality productElimination impliesFunctionality pointwiseFunctionality equalityTransitivity equalitySymmetry promote_hyp

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[a,b:\mBbbZ{}].    ((\mSigma{}(a  +  (b  *  i)  |  i  <  n)  *  2)  =  (n  *  (a  +  a  +  (b  *  (n  -  1)))))

Date html generated: 2017_04_14-AM-09_20_40
Last ObjectModification: 2017_02_27-PM-03_57_00

Theory : int_2

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