Nuprl Lemma : rminimum-positive
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. (r0 < rminimum(n;m;i.x[i]) ⇐⇒ ∀i:{n..m + 1-}. (r0 < x[i])) supposing n ≤ m
Proof
Definitions occuring in Statement :
rminimum: rminimum(n;m;k.x[k]),
rless: x < y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
iff: 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,
cand: A c∧ B,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
bfalse: ff,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
sq_type: SQType(T),
ge: i ≥ j ,
nat: ℕ,
rminimum: rminimum(n;m;k.x[k]),
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
guard: {T},
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
nat_plus: ℕ+,
sq_stable: SqStable(P),
real: ℝ,
subtype_rel: A ⊆r B,
sq_exists: ∃x:A [B[x]],
rless: x < y,
squash: ↓T,
less_than: a < b,
lelt: i ≤ j < k,
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
int_seg: {i..j-},
rev_implies: P ⇐ Q,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
le: A ≤ B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
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,
istype-false,
le_reflexive,
int_seg_subtype,
subtype_rel_function,
rmin_strict_ub,
all_wf,
primrec-wf2,
rmin_wf,
decidable__lt,
primrec_wf,
istype-less_than,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
primrec-unroll,
primrec0_lemma,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
nat_properties,
int_subtype_base,
subtype_base_sq,
int_term_value_subtract_lemma,
subtract_wf,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
itermSubtract_wf,
rless_transitivity1,
rleq_weakening,
rleq_weakening_equal,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermConstant_wf,
itermAdd_wf,
intformless_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_plus_properties,
sq_stable__less_than,
int_seg_properties,
rleq_functionality_wrt_implies,
rminimum_lb,
istype-int,
istype-le,
real_wf,
rminimum_wf,
int-to-real_wf,
rless_wf,
int_seg_wf,
le_witness_for_triv
Rules used in proof :
multiplyEquality,
minusEquality,
closedConclusion,
functionEquality,
setIsType,
productIsType,
promote_hyp,
equalityIstype,
equalityElimination,
applyLambdaEquality,
intEquality,
cumulativity,
instantiate,
dependent_set_memberEquality_alt,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
dependent_functionElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
imageElimination,
because_Cache,
setElimination,
inhabitedIsType,
functionIsType,
applyEquality,
lambdaEquality_alt,
sqequalRule,
natural_numberEquality,
addEquality,
universeIsType,
independent_pairFormation,
rename,
hypothesis,
independent_isectElimination,
hypothesisEquality,
productElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n,m:\mBbbZ{}.
\mforall{}x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}. (r0 < rminimum(n;m;i.x[i]) \mLeftarrow{}{}\mRightarrow{} \mforall{}i:\{n..m + 1\msupminus{}\}. (r0 < x[i])) supposing n \mleq{} m
Date html generated:
2019_11_06-PM-00_31_33
Last ObjectModification:
2019_11_05-PM-03_27_18
Theory : reals
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