Nuprl Lemma : rroot-abs-property
∀i:{2...}. ∀x:ℝ. (rroot-abs(i;x)^i = |x|)
Proof
Definitions occuring in Statement :
rroot-abs: rroot-abs(i;x),
rabs: |x|,
rnexp: x^k1,
req: x = y,
real: ℝ,
int_upper: {i...},
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
int_upper: {i...},
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
nat_plus: ℕ+,
int_nzero: ℤ-o,
true: True,
sq_type: SQType(T),
guard: {T},
so_lambda: λ2x.t[x],
real: ℝ,
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
rnexp: x^k1,
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
reg-seq-nexp: reg-seq-nexp(x;k),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bdd-diff: bdd-diff(f;g),
ge: i ≥ j ,
rroot-abs: rroot-abs(i;x),
squash: ↓T,
sq_stable: SqStable(P),
regular-int-seq: k-regular-seq(f),
less_than: a < b,
label: ...$L... t,
rev_uimplies: rev_uimplies(P;Q),
subtract: n - m,
cand: A c∧ B
Lemmas referenced :
canon-bnd_wf,
rroot-abs_wf,
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,
divide_wfa,
exp_wf2,
exp_wf3,
subtype_base_sq,
int_subtype_base,
nequal_wf,
add_nat_plus,
multiply_nat_wf,
add_nat_wf,
divide_wf,
exp_wf4,
subtype_rel_set,
int_upper_wf,
nat_wf,
nat_plus_wf,
le_wf,
absval_wf,
istype-int_upper,
upper_subtype_nat,
istype-false,
exp_wf_nat_plus,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
nat_plus_properties,
add-is-int-iff,
multiply-is-int-iff,
itermAdd_wf,
itermMultiply_wf,
intformeq_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
false_wf,
req-iff-bdd-diff,
rnexp_wf,
rabs_wf,
value-type-has-value,
set-value-type,
int-value-type,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
canonical-bound_wf,
real_wf,
accelerate_wf,
reg-seq-nexp_wf,
bdd-diff_functionality,
accelerate-bdd-diff,
bdd-diff_weakening,
exp-fastexp,
le_witness_for_triv,
nat_properties,
rabs-approx,
exp_preserves_lt,
nat_plus_subtype_nat,
less_than_wf,
squash_wf,
true_wf,
exp-zero,
subtype_rel_self,
iff_weakening_equal,
fastexp_wf,
iroot_wf,
mul_bounds_1a,
sq_stable__regular-int-seq,
iroot-property,
exp-positive,
absval_unfold,
lt_int_wf,
assert_of_lt_int,
istype-top,
iff_weakening_uiff,
assert_wf,
mul_cancel_in_le,
absval_mul,
exp_step,
set_subtype_base,
decidable__equal_int,
int_nzero_wf,
exp-of-mul,
istype-nat,
mul_preserves_le,
le_functionality,
le_weakening,
int-triangle-inequality,
mul_nzero,
nat_plus_inc_int_nzero,
mul-associates,
mul-distributes,
minus-one-mul,
mul-swap,
mul-commutes,
add-commutes,
add-associates,
add-swap,
add-mul-special,
zero-mul,
zero-add,
absval_nat_plus,
one-mul,
div_rem_sum2,
rem_bounds_absval,
sq_stable__less_than,
itermMinus_wf,
int_term_value_minus_lemma,
add_functionality_wrt_le,
exp-difference-inequality,
not-lt-2,
not-equal-2,
le-add-cancel,
subtract-is-int-iff,
exp-one,
mul_bounds_1b,
mul_nat_plus,
exp_preserves_le,
multiply_functionality_wrt_le,
absval_pos,
absval_sym,
sq_stable__le,
le_weakening2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
dependent_set_memberEquality_alt,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
addEquality,
multiplyEquality,
lambdaFormation_alt,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
baseClosed,
sqequalBase,
functionEquality,
inhabitedIsType,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
productElimination,
callbyvalueReduce,
equalityElimination,
setEquality,
functionIsType,
imageElimination,
imageMemberEquality,
universeEquality,
productIsType,
minusEquality,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isectIsTypeImplies,
divideEquality,
equalityIsType4,
remainderEquality,
hyp_replacement,
equalityIsType1
Latex:
\mforall{}i:\{2...\}. \mforall{}x:\mBbbR{}. (rroot-abs(i;x)\^{}i = |x|)
Date html generated:
2019_10_30-AM-07_56_17
Last ObjectModification:
2019_10_10-AM-10_22_50
Theory : reals
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