Nuprl Lemma : singleton_support_sum

`∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. ∀[m:ℕn].  Σ(f[x] | x < n) = f[m] ∈ ℤ supposing ∀x:ℕn. ((¬(x = m ∈ ℤ)) `` (f[x] = 0 ∈ ℤ))`

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` int_seg: `{i..j-}` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` prop: `ℙ` nat: `ℕ` so_lambda: `λ2x.t[x]` implies: `P `` Q` int_seg: `{i..j-}` so_apply: `x[s]` all: `∀x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` squash: `↓T` nequal: `a ≠ b ∈ T ` true: `True` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` ge: `i ≥ j ` lelt: `i ≤ j < k` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` not: `¬A` top: `Top`
Lemmas referenced :  all_wf int_seg_wf not_wf equal_wf equal-wf-T-base nat_wf isolate_summand empty_support ifthenelse_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int squash_wf true_wf iff_weakening_equal int_seg_properties nat_properties decidable__equal_int add-is-int-iff satisfiable-full-omega-tt intformand_wf intformnot_wf intformeq_wf itermVar_wf itermConstant_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_term_value_add_lemma int_formula_prop_wf false_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality sqequalRule lambdaEquality functionEquality intEquality applyEquality functionExtensionality baseClosed because_Cache isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry independent_isectElimination lambdaFormation unionElimination equalityElimination productElimination dependent_pairFormation promote_hyp dependent_functionElimination instantiate cumulativity independent_functionElimination voidElimination imageElimination universeEquality imageMemberEquality pointwiseFunctionality baseApply closedConclusion int_eqEquality voidEquality independent_pairFormation computeAll

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].  \mforall{}[m:\mBbbN{}n].    \mSigma{}(f[x]  |  x  <  n)  =  f[m]  supposing  \mforall{}x:\mBbbN{}n.  ((\mneg{}(x  =  m))  {}\mRightarrow{}  (f[x]  =  0))

Date html generated: 2017_04_14-AM-09_21_10
Last ObjectModification: 2017_02_27-PM-03_57_04

Theory : int_2

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