Nuprl Lemma : rmaximum_ub
∀[k,n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (x[k] ≤ rmaximum(n;m;i.x[i])) supposing ((k ≤ m) and (n ≤ k))
Proof
Definitions occuring in Statement :
rmaximum: rmaximum(n;m;k.x[k]),
rleq: x ≤ y,
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
true: True,
less_than': less_than'(a;b),
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
sq_type: SQType(T),
ge: i ≥ j ,
guard: {T},
nat: ℕ,
rmaximum: rmaximum(n;m;k.x[k]),
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
nat_plus: ℕ+,
false: False,
not: ¬A,
and: P ∧ Q,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
implies: P ⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rmax_functionality_wrt_rleq,
rleq_functionality_wrt_implies,
rleq-rmax,
subtract-add-cancel,
subtype_rel_self,
le-add-cancel,
add-commutes,
add-zero,
zero-add,
zero-mul,
add-mul-special,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
add-associates,
condition-implies-le,
not-le-2,
false_wf,
le_reflexive,
int_seg_subtype,
subtype_rel_function,
assert_of_le_int,
bnot_of_lt_int,
assert_functionality_wrt_uiff,
eqff_to_assert,
bnot_wf,
le_int_wf,
assert_of_lt_int,
eqtt_to_assert,
assert_wf,
equal-wf-base,
uiff_transitivity,
bool_wf,
lt_int_wf,
primrec-unroll,
rleq_weakening_equal,
equal_wf,
primrec0_lemma,
int_seg_properties,
rmax_wf,
primrec_wf,
less_than_wf,
ge_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
nat_properties,
int_subtype_base,
subtype_base_sq,
nat_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
real_wf,
le_wf,
nat_plus_wf,
lelt_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermConstant_wf,
itermAdd_wf,
intformless_wf,
decidable__lt,
int_seg_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
nat_plus_properties,
rmaximum_wf,
rsub_wf,
less_than'_wf,
decidable__le
Rules used in proof :
multiplyEquality,
baseClosed,
closedConclusion,
baseApply,
equalityElimination,
intWeakElimination,
applyLambdaEquality,
cumulativity,
instantiate,
lambdaFormation,
functionEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
dependent_set_memberEquality,
addEquality,
functionExtensionality,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
natural_numberEquality,
rename,
setElimination,
independent_isectElimination,
applyEquality,
isectElimination,
because_Cache,
independent_pairEquality,
productElimination,
lambdaEquality,
sqequalRule,
independent_functionElimination,
unionElimination,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
thin,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k,n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. (x[k] \mleq{} rmaximum(n;m;i.x[i])) supposing ((k \mleq{} m) and (n \mleq{} k))
Date html generated:
2018_05_22-PM-01_57_17
Last ObjectModification:
2018_05_21-AM-00_14_13
Theory : reals
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