Nuprl Lemma : seq-min-upper-property
∀[k,n:ℕ]. ∀[f:ℕ+ ⟶ ℤ].
∀i:ℕ+n + 1. (((i * (f seq-min-upper(k;n;f))) - seq-min-upper(k;n;f) * (f i)) ≤ ((2 * k) * (seq-min-upper(k;n;f) - i)))
Proof
Definitions occuring in Statement :
seq-min-upper: seq-min-upper(k;n;f),
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
less_than: a < b,
so_apply: x[s],
so_lambda: λ2x.t[x],
bnot: ¬bb,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
seq-min-upper: seq-min-upper(k;n;f),
or: P ∨ Q,
decidable: Dec(P),
true: True,
less_than': less_than'(a;b),
subtract: n - m,
uiff: uiff(P;Q),
squash: ↓T,
sq_stable: SqStable(P),
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
le: A ≤ B,
prop: ℙ,
and: P ∧ Q,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
subtract-is-int-iff,
mul_cancel_in_le,
mul_preserves_le,
set_subtype_base,
mul_preserves_lt,
nat_plus_properties,
not_wf,
lelt_wf,
assert-bnot,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
int_term_value_mul_lemma,
itermMultiply_wf,
multiply-is-int-iff,
add-is-int-iff,
assert_of_le_int,
eqtt_to_assert,
int_formula_prop_eq_lemma,
intformeq_wf,
le_int_wf,
int_subtype_base,
decidable__equal_int,
primrec-unroll,
iff_wf,
assert_wf,
assert_of_lt_int,
subtract-add-cancel,
bfalse_wf,
lt_int_wf,
iff_imp_equal_bool,
bool_subtype_base,
bool_wf,
subtype_base_sq,
nat_wf,
le-add-cancel,
not-lt-2,
false_wf,
decidable__lt,
le_wf,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
decidable__le,
le-add-cancel2,
add-commutes,
add_functionality_wrt_le,
zero-add,
add-associates,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
less-iff-le,
sq_stable__le,
int_seg_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_plus_wf,
int_term_value_add_lemma,
itermAdd_wf,
int_seg_properties,
seq-min-upper_wf,
subtract_wf,
less_than'_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties
Rules used in proof :
applyLambdaEquality,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
equalityElimination,
impliesFunctionality,
addLevel,
cumulativity,
instantiate,
functionEquality,
unionElimination,
minusEquality,
imageElimination,
baseClosed,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
dependent_set_memberEquality,
applyEquality,
functionExtensionality,
addEquality,
because_Cache,
multiplyEquality,
independent_pairEquality,
productElimination,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
thin,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k,n:\mBbbN{}]. \mforall{}[f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}].
\mforall{}i:\mBbbN{}\msupplus{}n + 1
(((i * (f seq-min-upper(k;n;f))) - seq-min-upper(k;n;f) * (f i)) \mleq{} ((2 * k)
* (seq-min-upper(k;n;f) - i)))
Date html generated:
2018_05_22-PM-01_33_31
Last ObjectModification:
2018_05_21-AM-00_09_16
Theory : reals
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