Nuprl Lemma : rmax-nonneg
∀[x,y:ℝ]. rnonneg(rmax(x;y)) supposing rnonneg(x) ∨ rnonneg(y)
Proof
Definitions occuring in Statement :
rnonneg: rnonneg(x),
rmax: rmax(x;y),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
or: P ∨ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
or: P ∨ Q,
rnonneg2: rnonneg2(x),
exists: ∃x:A. B[x],
rmax: rmax(x;y),
squash: ↓T,
nat_plus: ℕ+,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_upper: {i...},
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),
bnot: ¬bb,
assert: ↑b,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
less_than'_wf,
rmax_wf,
real_wf,
nat_plus_wf,
or_wf,
rnonneg_wf,
le_wf,
squash_wf,
true_wf,
imax_unfold,
iff_weakening_equal,
le_int_wf,
less_than_transitivity1,
less_than_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,
int_upper_wf,
all_wf,
rnonneg2_wf,
rnonneg-iff,
mul_preserves_le,
nat_plus_subtype_nat,
int_upper_properties,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
mul_cancel_in_le,
multiply-is-int-iff,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
lambdaEquality,
productElimination,
independent_pairEquality,
because_Cache,
extract_by_obid,
isectElimination,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
lambdaFormation,
unionElimination,
dependent_pairFormation,
imageElimination,
intEquality,
multiplyEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
dependent_set_memberEquality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
addLevel,
impliesFunctionality,
orFunctionality,
orLevelFunctionality,
functionEquality,
int_eqEquality,
voidEquality,
independent_pairFormation,
computeAll,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[x,y:\mBbbR{}]. rnonneg(rmax(x;y)) supposing rnonneg(x) \mvee{} rnonneg(y)
Date html generated:
2017_10_03-AM-08_24_26
Last ObjectModification:
2017_07_28-AM-07_23_20
Theory : reals
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