Nuprl Lemma : sum_aux_wf

`∀[v,i,k:ℤ]. ∀[f:{i..k-} ⟶ ℤ].  sum_aux(k;v;i;x.f[x]) ∈ ℤ supposing i ≤ k`

Proof

Definitions occuring in Statement :  sum_aux: `sum_aux(k;v;i;x.f[x])` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` member: `t ∈ T` function: `x:A ⟶ B[x]` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` 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` sum_aux: `sum_aux(k;v;i;x.f[x])` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` less_than: `a < b` less_than': `less_than'(a;b)` true: `True` squash: `↓T` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` decidable: `Dec(P)` has-value: `(a)↓` so_apply: `x[s]` int_seg: `{i..j-}` lelt: `i ≤ j < k` so_lambda: `λ2x.t[x]`
Lemmas referenced :  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 int_seg_wf lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf itermAdd_wf int_term_value_add_lemma eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma nat_wf value-type-has-value int-value-type lelt_wf int_subtype_base decidable__equal_int intformeq_wf int_formula_prop_eq_lemma decidable__lt le_wf
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality equalityTransitivity equalitySymmetry functionEquality addEquality isect_memberFormation unionElimination equalityElimination productElimination lessCases sqequalAxiom because_Cache imageMemberEquality baseClosed imageElimination promote_hyp instantiate cumulativity callbyvalueReduce applyEquality functionExtensionality dependent_set_memberEquality

Latex:
\mforall{}[v,i,k:\mBbbZ{}].  \mforall{}[f:\{i..k\msupminus{}\}  {}\mrightarrow{}  \mBbbZ{}].    sum\_aux(k;v;i;x.f[x])  \mmember{}  \mBbbZ{}  supposing  i  \mleq{}  k

Date html generated: 2017_04_14-AM-09_19_31
Last ObjectModification: 2017_02_27-PM-03_56_17

Theory : int_2

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