Nuprl Lemma : rnexp_wf
∀[k:ℕ]. ∀[x:ℝ]. (x^k ∈ ℝ)
Proof
Definitions occuring in Statement :
rnexp: x^k1,
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rnexp: x^k1,
has-value: (a)↓,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
real: ℝ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
int_upper: {i...},
prop: ℙ,
reg-seq-nexp: reg-seq-nexp(x;k),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
nequal: a ≠ b ∈ T
Lemmas referenced :
value-type-has-value,
nat_wf,
set-value-type,
le_wf,
istype-int,
int-value-type,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int-to-real_wf,
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,
int_upper_wf,
nat_plus_wf,
absval_wf,
istype-int_upper,
canon-bnd_wf,
reg-seq-nexp_wf,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformle_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-less_than,
accelerate_wf,
real_wf,
istype-nat,
subtract_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-le,
exp_wf4,
subtype_rel_set,
upper_subtype_nat,
istype-false,
exp_wf_nat_plus,
nat_plus_properties,
add_nat_plus,
multiply_nat_wf,
add_nat_wf,
divide_wf,
add-is-int-iff,
multiply-is-int-iff,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
false_wf,
exp-fastexp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_isectElimination,
intEquality,
lambdaEquality_alt,
natural_numberEquality,
hypothesisEquality,
dependent_functionElimination,
setElimination,
rename,
unionElimination,
instantiate,
cumulativity,
because_Cache,
independent_functionElimination,
inhabitedIsType,
lambdaFormation_alt,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
voidElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
setEquality,
functionEquality,
applyEquality,
multiplyEquality,
dependent_set_memberEquality_alt,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
independent_pairFormation,
universeIsType,
axiomEquality,
isectIsTypeImplies,
applyLambdaEquality,
addEquality,
divideEquality,
baseClosed,
sqequalBase,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[x:\mBbbR{}]. (x\^{}k \mmember{} \mBbbR{})
Date html generated:
2019_10_29-AM-09_34_02
Last ObjectModification:
2019_01_31-PM-08_16_15
Theory : reals
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