Nuprl Lemma : Riemann-integral-additive
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} ].
∀c:ℝ. ∫ f[x] dx on [a, b] = (∫ f[x] dx on [a, c] + ∫ f[x] dx on [c, b]) supposing (a ≤ c) ∧ (c ≤ b)
Proof
Definitions occuring in Statement :
Riemann-integral: ∫ f[x] dx on [a, b],
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
req: x = y,
radd: a + b,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
and: P ∧ Q,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
squash: ↓T,
label: ...$L... t,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
cand: A c∧ B,
sq_stable: SqStable(P),
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
top: Top,
converges-to: lim n→∞.x[n] = y,
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
sq_exists: ∃x:{A| B[x]},
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
rneq: x ≠ y,
rless: x < y,
le: A ≤ B,
uiff: uiff(P;Q),
subtract: n - m,
rleq: x ≤ y,
rnonneg: rnonneg(x),
Riemann-sum: Riemann-sum(f;a;b;k),
let: let,
partition: partition(I),
partitions: partitions(I;p),
i-finite: i-finite(I),
rccint: [l, u],
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
bfalse: ff,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
lt_int: i <z j,
bool: 𝔹,
unit: Unit,
sq_type: SQType(T),
bnot: ¬bb,
last: last(L),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
i-length: |I|,
rsub: x - y,
real: ℝ,
uniform-partition: uniform-partition(I;k),
partition-mesh: partition-mesh(I;p),
mklist: mklist(n;f),
full-partition: full-partition(I;p),
frs-mesh: frs-mesh(p),
rmaximum: rmaximum(n;m;k.x[k]),
partition-choice: partition-choice(p),
partition-sum: S(f;p),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
nequal: a ≠ b ∈ T
Lemmas referenced :
i-member_wf,
rccint_wf,
real_wf,
ifun_wf,
squash_wf,
icompact_wf,
rfun_wf,
interval_wf,
eta_conv,
rccint-icompact,
rleq_transitivity,
iff_weakening_equal,
ifun_subtype_3,
rleq_weakening_equal,
Riemann-integral_wf,
rleq_wf,
set_wf,
sq_stable__rleq,
sq_stable__ifun,
sq_stable__req,
radd_wf,
req-iff-rabs-rleq,
nat_plus_wf,
member_rccint_lemma,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
subtype_rel_sets,
Riemann-sums-converge-to,
rfun_subtype_3,
mul_nat_plus,
less_than_wf,
partition-sums-converge,
rless-int-fractions2,
nat_properties,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformnot_wf,
intformless_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rdiv_wf,
rless-int,
intformand_wf,
int_formula_prop_and_lemma,
rless_wf,
int-to-real_wf,
sq_stable__all,
nat_wf,
le_wf,
rabs_wf,
rsub_wf,
Riemann-sum_wf,
subtype_rel_set,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than'_wf,
uniform-partition_wf,
partition_wf,
append_wf,
cons_wf,
partitions_wf,
equal_wf,
select_wf,
left-endpoint_wf,
right-endpoint_wf,
last_wf,
list_wf,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
length_wf,
sq_stable__frs-non-dec,
frs-non-dec-sorted-by,
sorted-by-append,
sorted-by-cons,
partition-point-member,
int_seg_wf,
l_all_cons,
int_seg_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
select-append,
subtype_rel_list,
top_wf,
list_ind_nil_lemma,
stuck-spread,
base_wf,
list_ind_cons_lemma,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
select-cons-hd,
subtract_wf,
le_weakening2,
non_neg_length,
length_cons,
length_append,
length-append,
itermAdd_wf,
int_term_value_add_lemma,
last_append,
add_nat_plus,
length_wf_nat,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
true_wf,
last_cons,
assert_elim,
null_wf3,
bfalse_wf,
btrue_neq_bfalse,
assert_wf,
partition-mesh_wf,
i-length_wf,
exists_wf,
all_wf,
rleq_functionality,
mesh-uniform-partition,
req_weakening,
r-archimedean-implies2,
nat_plus_subtype_nat,
rleq_functionality_wrt_implies,
rleq-int,
radd-preserves-rleq,
rminus_wf,
uiff_transitivity,
radd_comm,
radd_functionality,
radd-rminus-assoc,
radd-zero-both,
rmul_preserves_rleq,
rless_transitivity1,
rmul_wf,
req_wf,
rmul_preserves_rleq2,
rmul-rdiv-cancel2,
req_functionality,
req_inversion,
rmul-assoc,
rmul_functionality,
rmul_comm,
rmul-ac,
rmul-rdiv-cancel,
radd-ac,
radd-rminus-both,
imax_wf,
imax_nat,
imax_lb,
decidable__equal_int,
int_subtype_base,
primrec0_lemma,
primrec1_lemma,
rmax_lb,
mklist_length,
add-subtract-cancel,
frs-mesh_wf,
nil_wf,
iff_imp_equal_bool,
length_nil,
iff_wf,
append_assoc,
last-cons,
assert_of_null,
btrue_wf,
append_is_nil,
and_wf,
equal-wf-T-base,
last_singleton_append,
length-singleton,
rmaximum_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
rmaximum-split,
rmax_wf,
add-member-int_seg2,
lelt_wf,
select_append_front,
rmaximum-shift,
select_append_back,
rmaximum_functionality,
select_cons_tl,
imax_ub,
default-partition-choice_wf,
full-partition_wf,
full-partition-non-dec,
partition-choice_wf,
int_seg_subtype_nat,
le_int_wf,
assert_of_le_int,
select-cons,
subtype_rel_self,
frs-non-dec_wf,
rsum-split,
full-partition-point-member,
l_all_wf2,
l_member_wf,
set_subtype_base,
add_functionality_wrt_eq,
add_nat_wf,
rsum-split-shift,
rsum_wf,
rsum_functionality,
partition-sum_wf,
rneq-int,
int_entire_a,
equal-wf-base,
rabs_functionality,
rsub_functionality,
rminus-radd,
req_transitivity,
radd-assoc,
rmul-identity1,
rmul-distrib2,
rminus-as-rmul,
radd-int,
rmul-zero-both,
rabs-difference-symmetry,
r-triangle-inequality,
radd_functionality_wrt_rleq,
rleq-int-fractions,
rmul-int-rdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
dependent_set_memberEquality,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
setEquality,
because_Cache,
imageElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
universeEquality,
productEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
natural_numberEquality,
multiplyEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
inrFormation,
functionEquality,
addEquality,
minusEquality,
independent_pairEquality,
axiomEquality,
promote_hyp,
hypothesis_subsumption,
equalityElimination,
instantiate,
cumulativity,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
addLevel,
levelHypothesis,
existsFunctionality,
allFunctionality,
impliesFunctionality,
allLevelFunctionality,
impliesLevelFunctionality,
hyp_replacement,
inlFormation,
functionExtensionality
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} ].
\mforall{}c:\mBbbR{}
\mint{} f[x] dx on [a, b] = (\mint{} f[x] dx on [a, c] + \mint{} f[x] dx on [c, b]) supposing (a \mleq{} c) \mwedge{} (c \mleq{} b)
Date html generated:
2017_10_04-PM-10_15_19
Last ObjectModification:
2017_07_28-AM-08_47_43
Theory : reals_2
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