Nuprl Lemma : rsum_functionality

[n,m:ℤ]. ∀[x,y:{n..m 1-} ⟶ ℝ].  Σ{x[k] n≤k≤m} = Σ{y[k] n≤k≤m} supposing x[k] y[k] for k ∈ [n,m]


Proof




Definitions occuring in Statement :  pointwise-req: x[k] y[k] for k ∈ [n,m] rsum: Σ{x[k] n≤k≤m} req: y real: int_seg: {i..j-} uimplies: supposing a uall: [x:A]. B[x] so_apply: x[s] function: x:A ⟶ B[x] add: m natural_number: $n int:
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a rsum: Σ{x[k] n≤k≤m} so_lambda: λ2x.t[x] so_apply: x[s] implies:  Q prop: and: P ∧ Q int_seg: {i..j-} lelt: i ≤ j < k callbyvalueall: callbyvalueall has-value: (a)↓ has-valueall: has-valueall(a) cand: c∧ B top: Top subtype_rel: A ⊆B nat: all: x:A. B[x] le: A ≤ B less_than: a < b or: P ∨ Q sq_type: SQType(T) guard: {T} uiff: uiff(P;Q) ifthenelse: if then else fi  btrue: tt iff: ⇐⇒ Q not: ¬A rev_implies:  Q bfalse: ff pointwise-req: x[k] y[k] for k ∈ [n,m] decidable: Dec(P) satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False
Lemmas referenced :  int_term_value_subtract_lemma int_formula_prop_less_lemma itermSubtract_wf intformless_wf int_formula_prop_wf int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermConstant_wf itermAdd_wf itermVar_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le subtract_wf int_seg_properties select-from-upto assert_of_bnot iff_weakening_uiff iff_transitivity eqff_to_assert assert_of_lt_int eqtt_to_assert bool_subtype_base bool_wf subtype_base_sq bool_cases not_wf bnot_wf assert_wf lt_int_wf select-map lelt_wf top_wf subtype_rel_list length-from-upto length-map length_wf nat_wf length_wf_nat map-length radd-list_functionality valueall-type-real-list evalall-reduce from-upto_wf less_than_wf le_wf and_wf map_wf real-valueall-type list-valueall-type list_wf valueall-type-has-valueall int-value-type value-type-has-value real_wf pointwise-req_wf int_seg_wf rsum_wf req_witness
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule lambdaEquality applyEquality addEquality natural_numberEquality hypothesis independent_functionElimination isect_memberEquality because_Cache equalityTransitivity equalitySymmetry functionEquality intEquality independent_isectElimination setEquality callbyvalueReduce voidElimination voidEquality setElimination rename independent_pairFormation lambdaFormation dependent_set_memberEquality productElimination productEquality dependent_functionElimination unionElimination instantiate cumulativity impliesFunctionality dependent_pairFormation int_eqEquality computeAll

Latex:
\mforall{}[n,m:\mBbbZ{}].  \mforall{}[x,y:\{n..m  +  1\msupminus{}\}  {}\mrightarrow{}  \mBbbR{}].
    \mSigma{}\{x[k]  |  n\mleq{}k\mleq{}m\}  =  \mSigma{}\{y[k]  |  n\mleq{}k\mleq{}m\}  supposing  x[k]  =  y[k]  for  k  \mmember{}  [n,m]



Date html generated: 2016_05_18-AM-07_44_49
Last ObjectModification: 2016_01_17-AM-02_07_11

Theory : reals


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