Nuprl Lemma : rmin-positive
∀x,y:ℝ. ((rpositive(x) ∧ rpositive(y)) ⇒ rpositive(rmin(x;y)))
Proof
Definitions occuring in Statement :
rpositive: rpositive(x),
rmin: rmin(x;y),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
rmin: rmin(x;y),
implies: P ⇒ Q,
and: P ∧ Q,
rpositive2: rpositive2(x),
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
real: ℝ,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
uimplies: b supposing a,
guard: {T},
squash: ↓T,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
le: A ≤ B,
decidable: Dec(P),
subtract: n - m,
less_than': less_than'(a;b),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
nat: ℕ,
less_than: a < b
Lemmas referenced :
imax_nat_plus,
le_wf,
imax_wf,
nat_plus_wf,
all_wf,
imin_wf,
rpositive2_wf,
rpositive-iff,
rmin_wf,
rpositive_wf,
real_wf,
imax_lb,
squash_wf,
true_wf,
imax_unfold,
imin_unfold,
iff_weakening_equal,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
mul_preserves_le,
nat_plus_properties,
decidable__le,
decidable__lt,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
mul-commutes,
minus-one-mul-top,
add_functionality_wrt_le,
add-associates,
add-zero,
add-swap,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermMultiply_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
mul_preserves_lt,
multiply-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
minus-add,
minus-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
hypothesisEquality,
hypothesis,
sqequalRule,
setElimination,
rename,
lambdaEquality,
functionEquality,
multiplyEquality,
applyEquality,
productEquality,
addLevel,
impliesFunctionality,
independent_pairFormation,
independent_functionElimination,
andLevelFunctionality,
independent_isectElimination,
dependent_functionElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
intEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
unionElimination,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
dependent_set_memberEquality,
addEquality,
minusEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}x,y:\mBbbR{}. ((rpositive(x) \mwedge{} rpositive(y)) {}\mRightarrow{} rpositive(rmin(x;y)))
Date html generated:
2017_10_03-AM-08_24_33
Last ObjectModification:
2017_07_28-AM-07_23_23
Theory : reals
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