### Nuprl Lemma : summand-le-sum

`∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ].  ∀[i:ℕn]. (f[i] ≤ Σ(f[x] | x < n)) supposing ∀x:ℕn. (0 ≤ f[x])`

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]` le: `A ≤ B` all: `∀x:A. B[x]` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` prop: `ℙ` so_apply: `x[s]` le: `A ≤ B` and: `P ∧ Q` all: `∀x:A. B[x]` nat: `ℕ` int_seg: `{i..j-}` lelt: `i ≤ j < k` so_lambda: `λ2x.t[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` less_than': `less_than'(a;b)` false: `False` not: `¬A` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` less_than: `a < b` squash: `↓T` ge: `i ≥ j ` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  isolate_summand le_wf le_witness_for_triv int_seg_wf istype-le istype-int istype-nat sum_wf ifthenelse_wf eq_int_wf eqtt_to_assert assert_of_eq_int istype-false eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot neg_assert_of_eq_int int_seg_properties nat_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma istype-less_than add-is-int-iff itermAdd_wf int_term_value_add_lemma false_wf le_functionality le_weakening add_functionality_wrt_le non_neg_sum
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality hyp_replacement equalitySymmetry applyLambdaEquality applyEquality because_Cache sqequalRule isect_memberEquality_alt productElimination equalityTransitivity independent_isectElimination isectIsTypeImplies inhabitedIsType functionIsType universeIsType natural_numberEquality setElimination rename addEquality lambdaEquality_alt intEquality lambdaFormation_alt unionElimination equalityElimination independent_pairFormation dependent_pairFormation_alt equalityIstype promote_hyp dependent_functionElimination instantiate cumulativity independent_functionElimination voidElimination imageElimination dependent_set_memberEquality_alt approximateComputation int_eqEquality Error :memTop,  productIsType pointwiseFunctionality baseApply closedConclusion baseClosed

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].    \mforall{}[i:\mBbbN{}n].  (f[i]  \mleq{}  \mSigma{}(f[x]  |  x  <  n))  supposing  \mforall{}x:\mBbbN{}n.  (0  \mleq{}  f[x])

Date html generated: 2020_05_19-PM-09_41_36
Last ObjectModification: 2020_01_23-PM-00_39_10

Theory : int_2

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