Nuprl Lemma : rroot-abs-non-neg
∀i:{2...}. ∀x:ℝ. ∀n:ℕ+. (0 ≤ (rroot-abs(i;x) n))
Proof
Definitions occuring in Statement :
rroot-abs: rroot-abs(i;x),
real: ℝ,
int_upper: {i...},
nat_plus: ℕ+,
le: A ≤ B,
all: ∀x:A. B[x],
apply: f a,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
rroot-abs: rroot-abs(i;x),
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
int_upper: {i...},
nat_plus: ℕ+,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
less_than': less_than'(a;b),
has-value: (a)↓,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
true: True,
ge: i ≥ j ,
squash: ↓T,
real: ℝ
Lemmas referenced :
exp-fastexp,
subtract_wf,
nat_plus_properties,
int_upper_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_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_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
exp_wf4,
false_wf,
nat_wf,
value-type-has-value,
set-value-type,
int-value-type,
exp_preserves_lt,
decidable__lt,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
nat_plus_subtype_nat,
nat_properties,
intformless_wf,
int_formula_prop_less_lemma,
squash_wf,
true_wf,
exp-zero,
exp_wf2,
iff_weakening_equal,
fastexp_wf,
int_upper_subtype_nat,
nat_plus_wf,
rabs_wf,
real_wf,
zero-le-nat,
iroot_wf,
equal_wf,
int_upper_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
dependent_set_memberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
because_Cache,
callbyvalueReduce,
productElimination,
independent_functionElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}i:\{2...\}. \mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}\msupplus{}. (0 \mleq{} (rroot-abs(i;x) n))
Date html generated:
2017_10_03-AM-10_41_02
Last ObjectModification:
2017_07_28-AM-08_17_03
Theory : reals
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