Nuprl Lemma : mesh-uniform-partition
∀[I:Interval]. ∀[k:ℕ+]. (partition-mesh(I;uniform-partition(I;k)) = (|I|/r(k))) supposing icompact(I)
Proof
Definitions occuring in Statement :
uniform-partition: uniform-partition(I;k),
partition-mesh: partition-mesh(I;p),
icompact: icompact(I),
i-length: |I|,
interval: Interval,
rdiv: (x/y),
req: x = y,
int-to-real: r(n),
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
partition-mesh: partition-mesh(I;p),
frs-mesh: frs-mesh(p),
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
full-partition: full-partition(I;p),
subtype_rel: A ⊆r B,
partition: partition(I),
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
le: A ≤ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
uniform-partition: uniform-partition(I;k),
nat: ℕ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
icompact: icompact(I),
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
less_than: a < b,
subtract: n - m,
i-length: |I|
Lemmas referenced :
req_witness,
partition-mesh_wf,
uniform-partition_wf,
rdiv_wf,
i-length_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,
nat_plus_wf,
icompact_wf,
interval_wf,
lt_int_wf,
length_wf,
real_wf,
full-partition_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
less_than_wf,
length_of_cons_lemma,
length_nil,
non_neg_length,
nil_wf,
partition_wf,
length_cons,
right-endpoint_wf,
cons_wf,
append_wf,
length_append,
subtype_rel_set,
list_wf,
top_wf,
partitions_wf,
subtype_rel_list,
length-append,
length_of_nil_lemma,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
le_int_wf,
le_wf,
bnot_wf,
rmaximum-constant,
subtract_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
rsub_wf,
select_wf,
int_seg_properties,
int_seg_wf,
mklist_length,
subtract-add-cancel,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
rmul_preserves_req,
rmul_wf,
radd_wf,
rminus_wf,
rinv_wf2,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal_wf,
req_functionality,
real_term_polynomial,
itermMultiply_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
req_transitivity,
rmul_functionality,
req_weakening,
rmul-rinv,
left-endpoint_wf,
lelt_wf,
add-member-int_seg2,
add-subtract-cancel,
radd_functionality,
rminus_functionality,
rmul_comm,
uniform-partition-point,
req_wf,
req_inversion,
radd-int,
rsub_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
setElimination,
rename,
sqequalRule,
inrFormation,
dependent_functionElimination,
because_Cache,
productElimination,
independent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
baseClosed,
applyEquality,
dependent_set_memberEquality,
addEquality,
lambdaFormation,
equalityElimination
Latex:
\mforall{}[I:Interval]
\mforall{}[k:\mBbbN{}\msupplus{}]. (partition-mesh(I;uniform-partition(I;k)) = (|I|/r(k))) supposing icompact(I)
Date html generated:
2017_10_03-AM-09_44_24
Last ObjectModification:
2017_07_28-AM-07_58_20
Theory : reals
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