Nuprl Lemma : rroot-abs_wf
∀i:{2...}. ∀x:ℝ. (rroot-abs(i;x) ∈ ℝ)
Proof
Definitions occuring in Statement :
rroot-abs: rroot-abs(i;x),
real: ℝ,
int_upper: {i...},
all: ∀x:A. B[x],
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
rroot-abs: rroot-abs(i;x),
uall: ∀[x:A]. B[x],
nat: ℕ,
int_upper: {i...},
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
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],
real: ℝ,
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
true: True,
ge: i ≥ j ,
squash: ↓T,
rabs: |x|,
regular-int-seq: k-regular-seq(f),
sq_type: SQType(T),
cand: A c∧ B,
less_than: a < b,
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
exp-fastexp,
subtract_wf,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-le,
exp_wf4,
istype-false,
value-type-has-value,
nat_wf,
set-value-type,
le_wf,
int-value-type,
real_wf,
istype-int_upper,
exp_preserves_lt,
decidable__lt,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
istype-less_than,
nat_plus_subtype_nat,
nat_plus_properties,
nat_properties,
intformless_wf,
int_formula_prop_less_lemma,
less_than_wf,
squash_wf,
true_wf,
exp-zero,
exp_wf2,
upper_subtype_nat,
subtype_rel_self,
iff_weakening_equal,
fastexp_wf,
nat_plus_wf,
rabs_wf,
iroot_wf,
mul_bounds_1a,
absval_wf,
absval-non-neg,
rroot-regularity-lemma,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
equal_wf,
istype-universe,
zero_ann_a,
itermMultiply_wf,
int_term_value_mul_lemma,
mul-commutes,
zero-mul,
iroot-zero,
set_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
exp_wf_nat_plus,
not-equal-2,
add-associates,
add-zero,
condition-implies-le,
minus-add,
minus-zero,
iroot-positive,
mul_nat_plus,
le_functionality,
le_weakening,
multiply_functionality_wrt_le,
iroot-property,
multiply-is-int-iff,
false_wf,
le_transitivity,
exp-of-mul,
itermAdd_wf,
int_term_value_add_lemma,
mul_preserves_lt,
exp_step,
real-regular,
mul_preserves_le,
absval_mul,
istype-nat,
mul-associates,
add_functionality_wrt_eq,
absval_pos,
regular-int-seq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
because_Cache,
inhabitedIsType,
callbyvalueReduce,
intEquality,
closedConclusion,
equalityIsType1,
equalityTransitivity,
equalitySymmetry,
productElimination,
applyEquality,
applyLambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
multiplyEquality,
cumulativity,
inrFormation_alt,
equalityIsType4,
inlFormation_alt,
productIsType,
addEquality,
minusEquality,
promote_hyp,
pointwiseFunctionality,
baseApply,
hyp_replacement
Latex:
\mforall{}i:\{2...\}. \mforall{}x:\mBbbR{}. (rroot-abs(i;x) \mmember{} \mBbbR{})
Date html generated:
2019_10_30-AM-07_55_41
Last ObjectModification:
2018_11_08-PM-02_38_44
Theory : reals
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