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: uimplies: supposing a uall: [x:A]. B[x] so_apply: x[s] all: x:A. B[x] not: ¬A implies:  Q 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 prop: nat: so_lambda: λ2x.t[x] implies:  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 then else 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 ∈  true: True subtype_rel: A ⊆B iff: ⇐⇒ Q rev_implies:  Q ge: i ≥  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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