Nuprl Lemma : rational-inner-approx_wf
∀[x:ℕ+ ⟶ ℤ]. ∀[n:ℕ+]. (rational-inner-approx(x;n) ∈ ℝ)
Proof
Definitions occuring in Statement :
rational-inner-approx: rational-inner-approx(x;n),
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rational-inner-approx: rational-inner-approx(x;n),
has-value: (a)↓,
uimplies: b supposing a,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
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,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B
Lemmas referenced :
value-type-has-value,
int-value-type,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
istype-int,
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_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
subtract_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
le_wf,
istype-le,
int-rdiv_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
nequal_wf,
int-to-real_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
because_Cache,
applyEquality,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
inhabitedIsType,
lambdaFormation_alt,
equalityElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
addEquality,
closedConclusion,
baseApply,
baseClosed,
sqequalBase,
axiomEquality,
isectIsTypeImplies,
functionIsType
Latex:
\mforall{}[x:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. (rational-inner-approx(x;n) \mmember{} \mBbbR{})
Date html generated:
2019_10_29-AM-10_02_19
Last ObjectModification:
2019_06_17-AM-11_37_02
Theory : reals
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