Nuprl Lemma : Riemann-sum-alt-req
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:[a, b] ⟶ℝ].
((∀x,y:ℝ. ((x ∈ [a, b]) ⇒ (x = y) ⇒ ((f y) = (f x))))
⇒ (∀[k:ℕ+]. (Riemann-sum-alt(f;a;b;k) = Riemann-sum(f;a;b;k))))
Proof
Definitions occuring in Statement :
Riemann-sum-alt: Riemann-sum-alt(f;a;b;k),
Riemann-sum: Riemann-sum(f;a;b;k),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
req: x = y,
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
cand: A c∧ B,
rccint: [l, u],
i-member: r ∈ I,
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)↓,
Riemann-sum-alt: Riemann-sum-alt(f;a;b;k),
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
sq_stable: SqStable(P),
prop: ℙ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rfun: I ⟶ℝ,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
lelt: i ≤ j < k,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
rge: x ≥ y,
subtract: n - m,
less_than: a < b,
partition-sum: partition-sum(f;x;p),
default-partition-choice: default-partition-choice(p),
has-valueall: has-valueall(a),
callbyvalueall: callbyvalueall,
Riemann-sum: Riemann-sum(f;a;b;k),
sq_type: SQType(T),
nat: ℕ,
full-partition: full-partition(I;p),
uniform-partition: uniform-partition(I;k),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
cons: [a / b],
select: L[n],
true: True,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
rsub: x - y
Lemmas referenced :
rmul-distrib2,
rmul-identity1,
rminus-as-rmul,
radd-ac,
radd-assoc,
rminus-radd,
rmul-one-both,
rminus_functionality,
rminus_wf,
rsub-rdiv,
rsub_functionality,
add-member-int_seg2,
set_subtype_base,
minus-minus,
minus-add,
rsub-int,
rdiv_functionality,
select-append,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
mklist_select,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
squash_wf,
true_wf,
select_cons_tl,
append_wf,
mklist_wf,
cons_wf,
nil_wf,
length_append,
subtype_rel_list,
top_wf,
iff_weakening_equal,
length-singleton,
minus-zero,
add-zero,
radd-zero-both,
radd_comm,
rmul-zero-both,
rmul_preserves_req,
list-set-type2,
select_wf,
rsum_functionality,
full-partition-point-member,
length_of_cons_lemma,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
length-append,
mklist_length,
le_wf,
length_of_nil_lemma,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
valueall-type-has-valueall,
list_wf,
list-valueall-type,
real-valueall-type,
full-partition_wf,
uniform-partition_wf,
evalall-reduce,
valueall-type-real-list,
rsum_linearity2,
rsum'-rsum,
req_functionality,
rsum'_wf,
subtract-add-cancel,
lelt_wf,
rsum_wf,
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_transitivity,
radd_functionality_wrt_rleq,
rleq_functionality_wrt_implies,
radd_functionality,
rmul_preserves_rleq2,
rleq-int,
decidable__le,
less_than'_wf,
rleq_weakening_equal,
rmul_comm,
rmul-rdiv-cancel2,
req_weakening,
rleq_functionality,
uiff_transitivity,
rmul_preserves_rleq,
int_seg_properties,
intformle_wf,
int_formula_prop_le_lemma,
int_seg_wf,
rmul_wf,
radd_wf,
subtract_wf,
set_wf,
rfun_wf,
member_rccint_lemma,
req_wf,
rccint_wf,
i-member_wf,
all_wf,
req_witness,
sq_stable__req,
Riemann-sum-alt_wf,
rleq_wf,
Riemann-sum_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,
rleq_transitivity,
rleq_weakening,
req_inversion,
and_wf
Rules used in proof :
imageElimination,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
inrFormation,
baseClosed,
imageMemberEquality,
independent_pairFormation,
functionEquality,
natural_numberEquality,
lambdaEquality,
intEquality,
independent_isectElimination,
callbyvalueReduce,
sqequalRule,
productElimination,
dependent_functionElimination,
introduction,
independent_functionElimination,
because_Cache,
hypothesis,
dependent_set_memberEquality,
hypothesisEquality,
isectElimination,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalHypSubstitution,
lemma_by_obid,
cut,
rename,
thin,
setElimination,
isect_memberFormation,
lambdaFormation,
applyEquality,
multiplyEquality,
addEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
independent_pairEquality,
instantiate,
setEquality,
cumulativity,
universeEquality,
equalityEquality,
promote_hyp,
equalityElimination
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f:[a, b] {}\mrightarrow{}\mBbbR{}].
((\mforall{}x,y:\mBbbR{}. ((x \mmember{} [a, b]) {}\mRightarrow{} (x = y) {}\mRightarrow{} ((f y) = (f x))))
{}\mRightarrow{} (\mforall{}[k:\mBbbN{}\msupplus{}]. (Riemann-sum-alt(f;a;b;k) = Riemann-sum(f;a;b;k))))
Date html generated:
2016_05_18-AM-10_45_38
Last ObjectModification:
2016_01_17-AM-00_42_13
Theory : reals
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