Nuprl Lemma : sum_functionality

[n:ℕ]. ∀[f,g:ℕn ⟶ ℤ].  Σ(f[x] x < n) = Σ(g[x] x < n) ∈ ℤ supposing ∀i:ℕn. (f[i] g[i] ∈ ℤ)


Proof




Definitions occuring in Statement :  sum: Σ(f[x] x < k) int_seg: {i..j-} nat: uimplies: supposing a uall: [x:A]. B[x] so_apply: x[s] all: x:A. B[x] function: x:A ⟶ B[x] natural_number: $n int: equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a so_lambda: λ2x.t[x] so_apply: x[s] nat: all: x:A. B[x] implies:  Q false: False ge: i ≥  satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A top: Top and: P ∧ Q prop: decidable: Dec(P) or: P ∨ Q subtype_rel: A ⊆B le: A ≤ B less_than': less_than'(a;b) int_seg: {i..j-} lelt: i ≤ j < k less_than: a < b bool: 𝔹 unit: Unit it: btrue: tt uiff: uiff(P;Q) ifthenelse: if then else fi  bfalse: ff iff: ⇐⇒ Q rev_implies:  Q squash: T true: True guard: {T}
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 all_wf equal_wf primrec0_lemma decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma subtype_rel_dep_function int_seg_subtype false_wf subtype_rel_self decidable__lt lelt_wf 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 squash_wf true_wf add_functionality_wrt_eq iff_weakening_equal nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality lambdaEquality applyEquality functionExtensionality natural_numberEquality setElimination rename because_Cache hypothesis lambdaFormation intWeakElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality functionEquality unionElimination dependent_set_memberEquality productElimination equalityElimination baseApply closedConclusion baseClosed impliesFunctionality imageElimination equalityTransitivity equalitySymmetry addEquality imageMemberEquality universeEquality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f,g:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].    \mSigma{}(f[x]  |  x  <  n)  =  \mSigma{}(g[x]  |  x  <  n)  supposing  \mforall{}i:\mBbbN{}n.  (f[i]  =  g[i])



Date html generated: 2017_04_14-AM-09_20_13
Last ObjectModification: 2017_02_27-PM-03_56_06

Theory : int_2


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