### Nuprl Lemma : sum-nat-le-simple

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

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` int_seg: `{i..j-}` nat: `ℕ` 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`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` nat: `ℕ` all: `∀x:A. B[x]` guard: `{T}` uimplies: `b supposing a` int_seg: `{i..j-}` ge: `i ≥ j ` lelt: `i ≤ j < k` and: `P ∧ Q` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` le: `A ≤ B`
Lemmas referenced :  sum-nat-le sum_wf nat_wf int_seg_wf int_seg_properties nat_properties decidable__le le_wf full-omega-unsat intformnot_wf intformle_wf itermVar_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf less_than'_wf
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule lambdaEquality applyEquality setElimination rename natural_numberEquality lambdaFormation independent_isectElimination because_Cache productElimination dependent_functionElimination dependent_set_memberEquality functionExtensionality unionElimination approximateComputation independent_functionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairEquality axiomEquality equalityTransitivity equalitySymmetry functionEquality

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

Date html generated: 2018_05_21-PM-00_28_24
Last ObjectModification: 2018_05_19-AM-06_59_49

Theory : int_2

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