Nuprl Lemma : rpositive-radd2
∀x,y:ℝ. (rpositive(x) ⇒ rnonneg(y) ⇒ rpositive(x + y))
Proof
Definitions occuring in Statement :
rnonneg: rnonneg(x),
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,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
real: ℝ,
uimplies: b supposing a,
guard: {T},
rnonneg2: rnonneg2(x),
rpositive2: rpositive2(x),
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
nat: ℕ,
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_upper: {i...},
subtract: n - m,
gt: i > j
Lemmas referenced :
pos_mul_arg_bounds,
le-add-cancel,
add_functionality_wrt_le,
add-zero,
mul-distributes-right,
minus-add,
zero-add,
zero-mul,
mul-swap,
mul-commutes,
mul-associates,
mul-distributes,
add-swap,
minus-one-mul-top,
add-commutes,
minus-one-mul,
add-associates,
condition-implies-le,
less-iff-le,
not-lt-2,
imax_ub,
int_term_value_add_lemma,
itermAdd_wf,
multiply-is-int-iff,
decidable__lt,
all_wf,
imax_wf,
le_wf,
false_wf,
mul_preserves_le,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
imax_lb,
imax_nat_plus,
rpositive_wf,
rnonneg_wf,
radd_wf,
rnonneg-iff,
rpositive-iff,
rnonneg2_wf,
mul_nat_plus,
l_sum_nil_lemma,
l_sum_cons_lemma,
map_nil_lemma,
map_cons_lemma,
iff_weakening_equal,
reg-seq-list-add-as-l_sum,
true_wf,
squash_wf,
rpositive2_wf,
accelerate-bdd-diff,
length_wf,
regular-int-seq_wf,
nat_plus_wf,
length_of_nil_lemma,
length_of_cons_lemma,
nil_wf,
real_wf,
cons_wf,
reg-seq-list-add_wf,
less_than_wf,
accelerate_wf,
rpositive2_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
introduction,
imageMemberEquality,
hypothesisEquality,
baseClosed,
hypothesis,
because_Cache,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
setEquality,
functionEquality,
intEquality,
setElimination,
rename,
independent_functionElimination,
productElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_isectElimination,
addLevel,
impliesFunctionality,
dependent_pairFormation,
multiplyEquality,
unionElimination,
int_eqEquality,
computeAll,
addEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
minusEquality,
inlFormation
Latex:
\mforall{}x,y:\mBbbR{}. (rpositive(x) {}\mRightarrow{} rnonneg(y) {}\mRightarrow{} rpositive(x + y))
Date html generated:
2016_05_18-AM-07_02_11
Last ObjectModification:
2016_01_17-AM-01_52_28
Theory : reals
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