Nuprl Lemma : adjacent-frs-points
∀[p:ℝ List]. ∀[i:ℕ||p|| - 1]. (frs-non-dec(p) ⇒ r0≤p[i + 1] - p[i]≤frs-mesh(p))
Proof
Definitions occuring in Statement :
frs-mesh: frs-mesh(p),
frs-non-dec: frs-non-dec(L),
rbetween: x≤y≤z,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
rbetween: x≤y≤z,
and: P ∧ Q,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
false: False,
int_seg: {i..j-},
uimplies: b supposing a,
nat_plus: ℕ+,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
frs-non-dec: frs-non-dec(L),
subtract: n - m,
rsub: x - y,
frs-mesh: frs-mesh(p),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
so_apply: x[s]
Lemmas referenced :
frs-non-dec_wf,
less_than'_wf,
rsub_wf,
select_wf,
real_wf,
nat_plus_properties,
int_seg_properties,
subtract_wf,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
subtract-is-int-iff,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
false_wf,
int-to-real_wf,
nat_plus_wf,
frs-mesh_wf,
int_seg_wf,
list_wf,
radd-preserves-rleq,
rleq_wf,
radd_wf,
rminus_wf,
lelt_wf,
add-member-int_seg2,
uiff_transitivity,
rleq_functionality,
radd_comm,
radd-ac,
req_weakening,
radd_functionality,
radd-rminus-both,
radd-zero-both,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
rmaximum_ub
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
because_Cache,
applyEquality,
addEquality,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
minusEquality,
axiomEquality,
dependent_set_memberEquality,
independent_functionElimination,
equalityElimination,
instantiate,
cumulativity
Latex:
\mforall{}[p:\mBbbR{} List]. \mforall{}[i:\mBbbN{}||p|| - 1]. (frs-non-dec(p) {}\mRightarrow{} r0\mleq{}p[i + 1] - p[i]\mleq{}frs-mesh(p))
Date html generated:
2017_10_03-AM-09_36_22
Last ObjectModification:
2017_07_28-AM-07_54_02
Theory : reals
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