### Nuprl Lemma : rng_sum-int

`∀[a,b:ℤ]. ∀[f:{a..b-} ⟶ ℤ].  (Σ(ℤ-rng) a ≤ i < b. f[i]) = Σ(f[a + i] | i < b - a) ∈ ℤ supposing a ≤ b`

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` function: `x:A ⟶ B[x]` subtract: `n - m` add: `n + m` int: `ℤ` equal: `s = t ∈ T` int_ring: `ℤ-rng` rng_sum: rng_sum
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` rng_sum: rng_sum mon_itop: `Π lb ≤ i < ub. E[i]` add_grp_of_rng: `r↓+gp` grp_op: `*` pi2: `snd(t)` pi1: `fst(t)` grp_id: `e` int_ring: `ℤ-rng` rng_plus: `+r` rng_zero: `0` nat: `ℕ` all: `∀x:A. B[x]` 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` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_seg: `{i..j-}` uiff: `uiff(P;Q)` lelt: `i ≤ j < k` ge: `i ≥ j ` itop: `Π(op,id) lb ≤ i < ub. E[i]` ycomb: `Y` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` infix_ap: `x f y` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` true: `True` le: `A ≤ B` subtract: `n - m` subtype_rel: `A ⊆r B` squash: `↓T` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  sum-as-primrec decidable__le subtract_wf satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_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_var_lemma int_formula_prop_wf le_wf int_seg_wf add-member-int_seg1 lelt_wf nat_properties intformless_wf int_formula_prop_less_lemma ge_wf less_than_wf lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int itermAdd_wf int_term_value_add_lemma eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot primrec-unroll eq_int_wf assert_of_eq_int neg_assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma nat_wf itop_wf decidable__lt primrec_wf decidable__equal_int add-associates minus-one-mul add-swap minus-one-mul-top add-commutes add-mul-special zero-mul zero-add itermMinus_wf int_term_value_minus_lemma add_functionality_wrt_eq iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin dependent_set_memberEquality because_Cache dependent_functionElimination natural_numberEquality hypothesisEquality hypothesis unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll applyEquality functionExtensionality setElimination rename productElimination lambdaFormation intWeakElimination independent_functionElimination axiomEquality functionEquality addEquality equalityElimination equalityTransitivity equalitySymmetry promote_hyp instantiate cumulativity minusEquality imageElimination imageMemberEquality baseClosed

Latex:
\mforall{}[a,b:\mBbbZ{}].  \mforall{}[f:\{a..b\msupminus{}\}  {}\mrightarrow{}  \mBbbZ{}].    (\mSigma{}(\mBbbZ{}-rng)  a  \mleq{}  i  <  b.  f[i])  =  \mSigma{}(f[a  +  i]  |  i  <  b  -  a)  supposing  a  \mleq{}  b

Date html generated: 2018_05_21-PM-08_27_23
Last ObjectModification: 2017_07_26-PM-05_54_57

Theory : general

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