Nuprl Lemma : rminimum-glb
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. ∀r:ℝ. (∀k:{n..m + 1-}. (r ≤ x[k])) ⇒ (r ≤ rminimum(n;m;i.x[i])) supposing n ≤ m
Proof
Definitions occuring in Statement :
rminimum: rminimum(n;m;k.x[k]),
rleq: x ≤ y,
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
true: True,
less_than': less_than'(a;b),
subtract: n - m,
subtype_rel: A ⊆r B,
cand: A c∧ B,
squash: ↓T,
less_than: a < b,
lelt: i ≤ j < k,
int_seg: {i..j-},
decidable: Dec(P),
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
prop: ℙ,
and: P ∧ Q,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
false: False,
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
rminimum: rminimum(n;m;k.x[k])
Lemmas referenced :
subtract-add-cancel,
subtype_rel_self,
le-add-cancel,
add-commutes,
add-associates,
add-zero,
zero-mul,
add-mul-special,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
not-le-2,
istype-false,
le_reflexive,
int_seg_subtype,
subtype_rel_function,
istype-nat,
int_seg_properties,
rmin_wf,
int_term_value_add_lemma,
itermAdd_wf,
decidable__lt,
decidable__le,
real_wf,
primrec_wf,
rmin_ub,
subtract-1-ge-0,
istype-le,
rleq_wf,
int_seg_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
intformnot_wf,
itermSubtract_wf,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
subtract_wf,
lt_int_wf,
primrec-unroll,
le_witness_for_triv,
istype-less_than,
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,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties
Rules used in proof :
multiplyEquality,
minusEquality,
imageElimination,
productIsType,
dependent_set_memberEquality_alt,
applyEquality,
addEquality,
functionIsType,
cumulativity,
instantiate,
promote_hyp,
equalityIstype,
equalityElimination,
unionElimination,
because_Cache,
inhabitedIsType,
functionIsTypeImplies,
equalitySymmetry,
equalityTransitivity,
productElimination,
universeIsType,
independent_pairFormation,
voidElimination,
isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
lambdaFormation_alt,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}n,m:\mBbbZ{}. \mforall{}x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}. \mforall{}r:\mBbbR{}.
(\mforall{}k:\{n..m + 1\msupminus{}\}. (r \mleq{} x[k])) {}\mRightarrow{} (r \mleq{} rminimum(n;m;i.x[i])) supposing n \mleq{} m
Date html generated:
2019_11_06-PM-00_30_54
Last ObjectModification:
2019_11_05-PM-00_15_25
Theory : reals
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