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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