Nuprl Lemma : rpositive-radd
∀x,y:ℝ. (rpositive(x) ⇒ rpositive(y) ⇒ rpositive(x + y))
Proof
Definitions occuring in Statement :
rpositive: rpositive(x),
radd: a + b,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
radd: a + b,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
top: Top,
real: ℝ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
guard: {T},
rpositive2: rpositive2(x),
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
nat: ℕ,
le: A ≤ B
Lemmas referenced :
rpositive2_functionality,
accelerate_wf,
less_than_wf,
reg-seq-list-add_wf,
cons_wf,
real_wf,
nil_wf,
length_of_cons_lemma,
length_of_nil_lemma,
nat_plus_wf,
regular-int-seq_wf,
length_wf,
accelerate-bdd-diff,
rpositive2_wf,
squash_wf,
true_wf,
reg-seq-list-add-as-l_sum,
iff_weakening_equal,
map_cons_lemma,
map_nil_lemma,
l_sum_cons_lemma,
l_sum_nil_lemma,
rpositive-iff,
radd_wf,
rpositive_wf,
imax_wf,
imax_nat_plus,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_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_eq_lemma,
int_formula_prop_wf,
equal_wf,
le_wf,
all_wf,
imax_lb,
right_mul_preserves_le,
imax_ub,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
mul_preserves_lt,
multiply-is-int-iff,
false_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
itermAdd_wf,
int_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
hypothesis,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
setEquality,
functionEquality,
intEquality,
functionExtensionality,
setElimination,
rename,
because_Cache,
independent_functionElimination,
productElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_isectElimination,
addLevel,
impliesFunctionality,
dependent_pairFormation,
applyLambdaEquality,
unionElimination,
int_eqEquality,
computeAll,
multiplyEquality,
addEquality,
inrFormation,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
inlFormation
Latex:
\mforall{}x,y:\mBbbR{}. (rpositive(x) {}\mRightarrow{} rpositive(y) {}\mRightarrow{} rpositive(x + y))
Date html generated:
2017_10_03-AM-08_23_20
Last ObjectModification:
2017_07_28-AM-07_22_49
Theory : reals
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