Nuprl Lemma : ireal-approx-radd
∀[x,y:ℝ]. ∀[j,i:ℕ]. ∀[M:ℕ+]. ∀[a,b:ℤ]. (j-approx(x;M;a) ⇒ i-approx(y;M;b) ⇒ j + i-approx(x + y;M;a + b))
Proof
Definitions occuring in Statement :
ireal-approx: j-approx(x;M;z),
radd: a + b,
real: ℝ,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
add: n + m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
ireal-approx: j-approx(x;M;z),
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
nat: ℕ,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rge: x ≥ y
Lemmas referenced :
ireal-approx_wf,
less_than'_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
rabs_wf,
radd_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
nat_plus_wf,
nat_wf,
real_wf,
rmul_preserves_req,
rmul_wf,
rinv_wf2,
rneq_functionality,
rmul-int,
req_weakening,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-T-base,
rminus_wf,
itermSubtract_wf,
itermAdd_wf,
req-iff-rsub-is-0,
minus-one-mul-top,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
itermMinus_wf,
rleq_wf,
rleq_weakening_equal,
req_functionality,
rsub_functionality,
rdiv_functionality,
req_inversion,
radd-int,
req_transitivity,
rmul_functionality,
rinv_functionality2,
rinv-of-rmul,
radd_functionality,
rmul-rinv3,
int-rinv-cancel,
squash_wf,
rminus-int,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_minus_lemma,
uimplies_transitivity,
rleq_functionality,
radd-rdiv,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
r-triangle-inequality,
rabs_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
because_Cache,
applyEquality,
addEquality,
setElimination,
rename,
independent_isectElimination,
inrFormation,
independent_functionElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
multiplyEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality,
addLevel,
instantiate,
cumulativity,
imageElimination
Latex:
\mforall{}[x,y:\mBbbR{}]. \mforall{}[j,i:\mBbbN{}]. \mforall{}[M:\mBbbN{}\msupplus{}]. \mforall{}[a,b:\mBbbZ{}].
(j-approx(x;M;a) {}\mRightarrow{} i-approx(y;M;b) {}\mRightarrow{} j + i-approx(x + y;M;a + b))
Date html generated:
2018_05_22-PM-01_59_25
Last ObjectModification:
2017_10_25-AM-10_40_07
Theory : reals
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