Nuprl Lemma : Riemann-sum-constant
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[c:ℝ]. ∀[k:ℕ+]. (Riemann-sum(λx.c;a;b;k) = (c * (b - a)))
Proof
Definitions occuring in Statement :
Riemann-sum: Riemann-sum(f;a;b;k),
rleq: x ≤ y,
rsub: x - y,
req: x = y,
rmul: a * b,
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
lambda: λx.A[x]
Definitions unfolded in proof :
squash: ↓T,
implies: P ⇒ Q,
sq_stable: SqStable(P),
uimplies: b supposing a,
and: P ∧ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
all: ∀x:A. B[x],
rfun: I ⟶ℝ,
subtype_rel: A ⊆r B,
prop: ℙ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
has-value: (a)↓,
Riemann-sum: Riemann-sum(f;a;b;k),
iff: P ⇐⇒ Q,
has-valueall: has-valueall(a),
callbyvalueall: callbyvalueall,
partition-sum: partition-sum(f;x;p),
default-partition-choice: default-partition-choice(p),
rev_uimplies: rev_uimplies(P;Q),
le: A ≤ B,
uiff: uiff(P;Q),
less_than: a < b,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
guard: {T},
int_seg: {i..j-},
cons: [a / b],
select: L[n],
rev_implies: P ⇐ Q,
int_upper: {i...},
true: True,
less_than': less_than'(a;b),
ge: i ≥ j ,
partition: partition(I),
nat: ℕ,
full-partition: full-partition(I;p),
sq_type: SQType(T)
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
select_append_back,
squash_wf,
true_wf,
lelt_wf,
select-cons-tl,
rsum-telescopes,
length_of_cons_lemma,
right_endpoint_rccint_lemma,
add_nat_wf,
append_wf,
cons_wf,
nil_wf,
length_nil,
non_neg_length,
length_cons,
partition_wf,
length_append,
subtype_rel_set,
partitions_wf,
subtype_rel_list,
length-append,
length_of_nil_lemma,
nat_wf,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
add_nat_plus,
length_wf_nat,
add_functionality_wrt_eq,
iff_weakening_equal,
length-singleton,
member_wf,
req_weakening,
left_endpoint_rccint_lemma,
req_wf,
req_functionality,
rsum_wf,
subtract_wf,
length_wf,
select_wf,
int_seg_properties,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
add-is-int-iff,
subtract-is-int-iff,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
false_wf,
int_seg_wf,
rsum_functionality2,
rmul-rsub-distrib,
le_wf,
full-partition_wf,
rccint_wf,
uniform-partition_wf,
list_wf,
valueall-type-has-valueall,
list-valueall-type,
real-valueall-type,
evalall-reduce,
valueall-type-real-list,
rccint-icompact,
value-type-has-value,
set-value-type,
less_than_wf,
int-value-type,
sq_stable__req,
Riemann-sum_wf,
rleq_wf,
top_wf,
member_rccint_lemma,
subtype_rel_dep_function,
real_wf,
and_wf,
subtype_rel_self,
set_wf,
rmul_wf,
rsub_wf,
req_witness,
nat_plus_wf
Rules used in proof :
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
lambdaFormation,
because_Cache,
independent_isectElimination,
setEquality,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
sqequalRule,
applyEquality,
lambdaEquality,
hypothesis,
dependent_set_memberEquality,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
rename,
thin,
setElimination,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
natural_numberEquality,
intEquality,
callbyvalueReduce,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityEquality,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
computeAll,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation,
unionElimination,
addEquality,
substitution,
universeEquality,
instantiate
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[c:\mBbbR{}]. \mforall{}[k:\mBbbN{}\msupplus{}]. (Riemann-sum(\mlambda{}x.c;a;b;k) = (c * (b - a)))
Date html generated:
2016_05_18-AM-10_39_54
Last ObjectModification:
2016_01_17-AM-00_24_47
Theory : reals
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