Nuprl Lemma : qmul_positive
∀[r,s:ℚ]. (↑qpositive(r * s)) supposing ((↑qpositive(s)) and (↑qpositive(r)))
Proof
Definitions occuring in Statement :
qpositive: qpositive(r),
qmul: r * s,
rationals: ℚ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
nat_plus: ℕ+,
cand: A c∧ B,
not: ¬A,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
qdiv: (r/s),
top: Top,
ifthenelse: if b then t else f fi ,
btrue: tt,
mk-rational: mk-rational(a;b),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
bfalse: ff,
or: P ∨ Q,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
q-elim,
nat_plus_properties,
assert-qeq,
int-subtype-rationals,
assert_wf,
qeq_wf2,
not_wf,
equal-wf-base,
rationals_wf,
int_subtype_base,
qinv-elim,
qmul-elim,
isint-int,
mk-rational_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
nequal_wf,
qpositive-elim,
mul_nzero,
mul-associates,
mul-commutes,
mul-swap,
one-mul,
mul_bounds_1b,
decidable__lt,
intformnot_wf,
intformor_wf,
int_formula_prop_not_lemma,
int_formula_prop_or_lemma,
less_than_wf,
or_wf,
iff_transitivity,
bor_wf,
band_wf,
lt_int_wf,
iff_weakening_uiff,
assert_of_bor,
assert_of_band,
assert_of_lt_int,
assert_witness,
isect_wf,
qpositive_wf,
qmul_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
isectElimination,
hypothesis,
setElimination,
rename,
addLevel,
impliesFunctionality,
applyEquality,
sqequalRule,
natural_numberEquality,
independent_isectElimination,
because_Cache,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
lambdaFormation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
multiplyEquality,
isintReduceTrue,
inlFormation,
unionElimination,
productEquality,
independent_functionElimination,
orFunctionality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
isectEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[r,s:\mBbbQ{}]. (\muparrow{}qpositive(r * s)) supposing ((\muparrow{}qpositive(s)) and (\muparrow{}qpositive(r)))
Date html generated:
2016_10_25-AM-11_51_13
Last ObjectModification:
2016_07_12-AM-07_48_07
Theory : rationals
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