Nuprl Lemma : uniform-partition-refines
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀k,m:ℕ+. uniform-partition([a, b];m * k) refines uniform-partition([a, b];k)
Proof
Definitions occuring in Statement :
partition-refines: P refines Q,
uniform-partition: uniform-partition(I;k),
rccint: [l, u],
rleq: x ≤ y,
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
multiply: n * m
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
squash: ↓T,
partition-refines: P refines Q,
frs-refines: frs-refines(p;q),
l_all: (∀x∈L.P[x]),
uniform-partition: uniform-partition(I;k),
top: Top,
nat: ℕ,
nat_plus: ℕ+,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
prop: ℙ,
int_seg: {i..j-},
i-finite: i-finite(I),
rccint: [l, u],
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
rneq: x ≠ y,
rev_implies: P ⇐ Q,
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
real: ℝ,
partition: partition(I),
so_lambda: λ2x.t[x],
so_apply: x[s],
l_exists: (∃x∈L. P[x]),
le: A ≤ B,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
less_than: a < b,
cand: A c∧ B,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2
Lemmas referenced :
rccint-icompact,
sq_stable__rleq,
mklist_length,
subtract_wf,
int_seg_properties,
length_wf,
nat_plus_wf,
mklist_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
rdiv_wf,
radd_wf,
rmul_wf,
rsub_wf,
int-to-real_wf,
left-endpoint_wf,
rccint_wf,
right-endpoint_wf,
rless-int,
decidable__lt,
rless_wf,
real_wf,
int_seg_wf,
uniform-partition_wf,
partition_wf,
set_wf,
rleq_wf,
mul_preserves_le,
itermAdd_wf,
int_term_value_add_lemma,
mul_preserves_lt,
itermMultiply_wf,
int_term_value_mul_lemma,
lelt_wf,
req_wf,
select_wf,
multiply-is-int-iff,
int_subtype_base,
false_wf,
not-lt-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
mul-distributes-right,
mul-commutes,
mul-associates,
one-mul,
less-iff-le,
mul-distributes,
zero-add,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
less_than_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
squash_wf,
true_wf,
mklist_select,
iff_weakening_equal,
rmul-is-positive,
subtract-add-cancel,
rmul_preserves_req,
rminus_wf,
rinv_wf2,
req_weakening,
req_functionality,
rdiv_functionality,
radd_functionality,
rmul_functionality,
req_inversion,
rmul-int,
rsub_functionality,
req_transitivity,
real_term_polynomial,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
rmul-rinv3,
rminus_functionality,
rinv-of-rmul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
productElimination,
independent_functionElimination,
isectElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
functionEquality,
intEquality,
because_Cache,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
independent_pairFormation,
computeAll,
addEquality,
inrFormation,
applyEquality,
multiplyEquality,
baseApply,
closedConclusion,
minusEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
inlFormation
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a \mleq{} b\} . \mforall{}k,m:\mBbbN{}\msupplus{}. uniform-partition([a, b];m * k) refines uniform-partition([a, b];k\000C)
Date html generated:
2017_10_03-AM-09_46_40
Last ObjectModification:
2017_07_28-AM-07_59_45
Theory : reals
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