### Nuprl Lemma : sum_le

`∀[k:ℕ]. ∀[f,g:ℕk ⟶ ℤ].  Σ(f[x] | x < k) ≤ Σ(g[x] | x < k) supposing ∀x:ℕk. (f[x] ≤ g[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` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat: `ℕ` all: `∀x:A. B[x]` implies: `P `` Q` false: `False` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` le: `A ≤ B` less_than': `less_than'(a;b)` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  sum-as-primrec int_seg_wf 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 less_than'_wf primrec_wf all_wf le_wf primrec0_lemma false_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma subtype_rel_dep_function int_seg_subtype subtype_rel_self decidable__lt lelt_wf primrec-unroll eq_int_wf bool_wf uiff_transitivity equal-wf-base int_subtype_base assert_wf eqtt_to_assert assert_of_eq_int iff_transitivity bnot_wf not_wf iff_weakening_uiff eqff_to_assert assert_of_bnot itermAdd_wf int_term_value_add_lemma equal_wf sum_wf nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality lambdaEquality applyEquality functionExtensionality natural_numberEquality setElimination rename because_Cache hypothesis lambdaFormation intWeakElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination productElimination independent_pairEquality addEquality axiomEquality equalityTransitivity equalitySymmetry functionEquality dependent_set_memberEquality unionElimination equalityElimination baseApply closedConclusion baseClosed impliesFunctionality

Latex:
\mforall{}[k:\mBbbN{}].  \mforall{}[f,g:\mBbbN{}k  {}\mrightarrow{}  \mBbbZ{}].    \mSigma{}(f[x]  |  x  <  k)  \mleq{}  \mSigma{}(g[x]  |  x  <  k)  supposing  \mforall{}x:\mBbbN{}k.  (f[x]  \mleq{}  g[x])

Date html generated: 2017_04_14-AM-09_20_23
Last ObjectModification: 2017_02_27-PM-03_56_58

Theory : int_2

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