Nuprl Lemma : uniform-partition-point
∀[I:Interval]
∀[k:ℕ+]
∀i:ℕk + 1
((full-partition(I;uniform-partition(I;k))[i] * r(k))
= (((r(k) - r(i)) * left-endpoint(I)) + (r(i) * right-endpoint(I))))
supposing icompact(I)
Proof
Definitions occuring in Statement :
uniform-partition: uniform-partition(I;k),
full-partition: full-partition(I;p),
icompact: icompact(I),
right-endpoint: right-endpoint(I),
left-endpoint: left-endpoint(I),
interval: Interval,
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
select: L[n],
int_seg: {i..j-},
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
uniform-partition: uniform-partition(I;k),
full-partition: full-partition(I;p),
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
select: L[n],
cons: [a / b],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
nat: ℕ,
icompact: icompact(I),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
true: True,
subtract: n - m,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
rat_term_to_real: rat_term_to_real(f;t),
rtermAdd: left "+" right,
rat_term_ind: rat_term_ind,
rtermMultiply: left "*" right,
rtermVar: rtermVar(var),
rtermSubtract: left "-" right,
pi1: fst(t),
rtermDivide: num "/" denom,
pi2: snd(t)
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_wf,
req_witness,
rmul_wf,
select_wf,
real_wf,
full-partition_wf,
uniform-partition_wf,
int_seg_properties,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
length_of_cons_lemma,
length-append,
mklist_length,
subtract_wf,
itermSubtract_wf,
intformless_wf,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
istype-le,
length_of_nil_lemma,
subtract-add-cancel,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
int-to-real_wf,
radd_wf,
rsub_wf,
left-endpoint_wf,
right-endpoint_wf,
nat_plus_wf,
icompact_wf,
interval_wf,
itermMultiply_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
rless-int,
intformeq_wf,
int_formula_prop_eq_lemma,
rless_wf,
append_wf,
mklist_wf,
rdiv_wf,
cons_wf,
nil_wf,
le_wf,
squash_wf,
true_wf,
length_append,
subtype_rel_list,
top_wf,
iff_weakening_equal,
length-singleton,
req_wf,
select_cons_tl,
subtype_rel_self,
length_wf,
istype-less_than,
select_append_back,
minus-add,
minus-minus,
add-associates,
minus-one-mul,
add-swap,
add-mul-special,
add-commutes,
zero-add,
zero-mul,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermAdd_wf,
rtermSubtract_wf,
rtermVar_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
select_append_front,
mklist_select
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
sqequalRule,
universeIsType,
natural_numberEquality,
addEquality,
lambdaEquality_alt,
imageElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
functionIsTypeImplies,
inhabitedIsType,
isectIsTypeImplies,
closedConclusion,
inrFormation_alt,
applyEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
functionIsType,
imageMemberEquality,
baseClosed,
universeEquality,
productIsType,
multiplyEquality,
equalityElimination,
int_eqReduceTrueSq,
equalityIstype,
promote_hyp,
int_eqReduceFalseSq
Latex:
\mforall{}[I:Interval]
\mforall{}[k:\mBbbN{}\msupplus{}]
\mforall{}i:\mBbbN{}k + 1
((full-partition(I;uniform-partition(I;k))[i] * r(k))
= (((r(k) - r(i)) * left-endpoint(I)) + (r(i) * right-endpoint(I))))
supposing icompact(I)
Date html generated:
2019_10_29-AM-10_49_22
Last ObjectModification:
2019_04_02-AM-09_55_19
Theory : reals
Home
Index