Nuprl Lemma : rmin-rnexp
∀[n:ℕ]. ∀[x,y:ℝ]. ((r0 ≤ x) ⇒ (r0 ≤ y) ⇒ (rmin(x^n;y^n) = rmin(x;y)^n))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rmin: rmin(x;y),
rnexp: x^k1,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uimplies: b supposing a,
prop: ℙ,
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
true: True,
subtract: n - m,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
rleq_antisymmetry,
rmin_wf,
rnexp_wf,
rleq_wf,
int-to-real_wf,
req_witness,
real_wf,
nat_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
less_than'_wf,
rsub_wf,
nat_plus_properties,
nat_plus_wf,
rnexp_zero_lemma,
le_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
rleq_weakening_equal,
rmin_lb,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rmul_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
rmin_ub,
rnexp-nonneg,
rleq_functionality,
rmin_functionality,
rnexp_unroll,
rleq_functionality_wrt_implies,
rmul_functionality_wrt_rleq2,
rmul_comm,
req_weakening,
rmul-rmin,
rmin-rleq,
not-rless,
rless_wf,
rmin_strict_ub,
not_wf,
rnexp-rleq-iff,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
zero-add,
add-zero,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
add-swap,
le-add-cancel2,
minus-minus,
minus-one-mul,
minus-one-mul-top,
rleq_weakening_rless,
rmul_preserves_rleq2,
sq_stable__less_than,
rless_transitivity1,
rless_irreflexivity,
rnexp-rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
natural_numberEquality,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
because_Cache,
setElimination,
rename,
intWeakElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
productElimination,
independent_pairEquality,
applyEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
unionElimination,
inlFormation,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
productEquality,
addLevel,
impliesFunctionality,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}]. ((r0 \mleq{} x) {}\mRightarrow{} (r0 \mleq{} y) {}\mRightarrow{} (rmin(x\^{}n;y\^{}n) = rmin(x;y)\^{}n))
Date html generated:
2017_10_03-AM-08_46_20
Last ObjectModification:
2017_07_28-AM-07_32_31
Theory : reals
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