Nuprl Lemma : Riemann-sum-alt_wf

`∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:[a, b] ⟶ℝ]. ∀[k:ℕ+].  (Riemann-sum-alt(f;a;b;k) ∈ ℝ)`

Proof

Definitions occuring in Statement :  Riemann-sum-alt: `Riemann-sum-alt(f;a;b;k)` rfun: `I ⟶ℝ` rccint: `[l, u]` rleq: `x ≤ y` real: `ℝ` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` member: `t ∈ T` set: `{x:A| B[x]} `
Definitions unfolded in proof :  prop: `ℙ` top: `Top` not: `¬A` false: `False` satisfiable_int_formula: `satisfiable_int_formula(fmla)` decidable: `Dec(P)` rev_implies: `P `` Q` or: `P ∨ Q` guard: `{T}` rneq: `x ≠ y` squash: `↓T` exists: `∃x:A. B[x]` real: `ℝ` so_apply: `x[s]` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` uimplies: `b supposing a` has-value: `(a)↓` implies: `P `` Q` and: `P ∧ Q` iff: `P `⇐⇒` Q` all: `∀x:A. B[x]` Riemann-sum-alt: `Riemann-sum-alt(f;a;b;k)` member: `t ∈ T` uall: `∀[x:A]. B[x]` rev_uimplies: `rev_uimplies(P;Q)` uiff: `uiff(P;Q)` lelt: `i ≤ j < k` int_seg: `{i..j-}` cand: `A c∧ B` rfun: `I ⟶ℝ` subtype_rel: `A ⊆r B` le: `A ≤ B` rnonneg: `rnonneg(x)` rleq: `x ≤ y` rge: `x ≥ y` subtract: `n - m`
Lemmas referenced :  itermSubtract_wf int_term_value_subtract_lemma zero-add zero-mul add-mul-special add-commutes add-swap minus-one-mul add-associates radd-int rmul_functionality rmul-distrib req_inversion req_transitivity radd_functionality_wrt_rleq rleq_weakening_equal rleq_functionality_wrt_implies radd_functionality rmul_preserves_rleq2 rleq-int decidable__le less_than'_wf rmul_comm rmul-rdiv-cancel2 req_weakening rleq_functionality uiff_transitivity rmul_wf rsum'_wf subtract_wf member_rccint_lemma rmul_preserves_rleq subtract-add-cancel int_seg_properties intformle_wf int_formula_prop_le_lemma and_wf int_seg_wf radd_wf rccint-icompact value-type-has-value nat_plus_wf set-value-type less_than_wf int-value-type real_wf regular-int-seq_wf function-value-type rdiv_wf rsub_wf int-to-real_wf rless-int nat_plus_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf rless_wf rfun_wf rccint_wf set_wf rleq_wf
Rules used in proof :  equalitySymmetry equalityTransitivity axiomEquality computeAll voidEquality voidElimination isect_memberEquality int_eqEquality dependent_pairFormation unionElimination inrFormation baseClosed imageMemberEquality independent_pairFormation because_Cache functionEquality natural_numberEquality lambdaEquality intEquality independent_isectElimination isectElimination callbyvalueReduce sqequalRule hypothesis independent_functionElimination productElimination hypothesisEquality dependent_functionElimination sqequalHypSubstitution lemma_by_obid rename thin setElimination cut introduction isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution addEquality dependent_set_memberEquality applyEquality minusEquality independent_pairEquality multiplyEquality

Latex:
\mforall{}[a:\mBbbR{}].  \mforall{}[b:\{b:\mBbbR{}|  a  \mleq{}  b\}  ].  \mforall{}[f:[a,  b]  {}\mrightarrow{}\mBbbR{}].  \mforall{}[k:\mBbbN{}\msupplus{}].    (Riemann-sum-alt(f;a;b;k)  \mmember{}  \mBbbR{})

Date html generated: 2016_05_18-AM-10_44_38
Last ObjectModification: 2016_01_17-AM-00_21_30

Theory : reals

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