Nuprl Lemma : rpositive-rmul
∀x,y:ℝ. (rpositive(x) ⇒ rpositive(y) ⇒ rpositive(x * y))
Proof
Definitions occuring in Statement :
rpositive: rpositive(x),
rmul: a * b,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rmul: a * b,
has-value: (a)↓,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
int_upper: {i...},
so_lambda: λ2x.t[x],
real: ℝ,
so_apply: x[s],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
rpositive2: rpositive2(x),
exists: ∃x:A. B[x],
nat: ℕ,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
cand: A c∧ B,
reg-seq-mul: reg-seq-mul(x;y),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
gt: i > j
Lemmas referenced :
real-has-value,
rpositive2_functionality,
accelerate_wf,
imax_wf,
canonical-bound_wf,
int_upper_wf,
all_wf,
nat_plus_wf,
le_wf,
absval_wf,
less_than_wf,
reg-seq-mul_wf2,
reg-seq-mul_wf,
accelerate-bdd-diff,
rpositive2_wf,
rpositive-iff,
rmul_wf,
rpositive_wf,
real_wf,
nat_wf,
ifthenelse_wf,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
add_nat_plus,
multiply_nat_wf,
subtype_rel_set,
int_upper_subtype_nat,
false_wf,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_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_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
add-is-int-iff,
multiply-is-int-iff,
squash_wf,
true_wf,
add_functionality_wrt_eq,
imax_unfold,
iff_weakening_equal,
imax_nat_plus,
multiply_nat_plus,
imax_ub,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
mul_nat_plus,
mul_preserves_le,
nat_plus_subtype_nat,
mul_preserves_lt,
mul_cancel_in_le,
equal-wf-base,
mul-swap,
div_rem_sum2,
nequal_wf,
left_mul_subtract_distrib,
rem_bounds_absval,
set_wf,
int_subtype_base,
decidable__equal_int,
sq_stable__less_than,
le_functionality,
le_weakening,
multiply_functionality_wrt_le,
sq_stable__le,
absval_pos,
neg_mul_arg_bounds,
gt_wf,
mul_bounds_1a,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
callbyvalueReduce,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
dependent_set_memberEquality,
addEquality,
multiplyEquality,
natural_numberEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
because_Cache,
independent_functionElimination,
productElimination,
addLevel,
impliesFunctionality,
functionEquality,
intEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
promote_hyp,
instantiate,
cumulativity,
pointwiseFunctionality,
baseApply,
closedConclusion,
imageElimination,
universeEquality,
inlFormation,
inrFormation,
productEquality,
divideEquality,
remainderEquality
Latex:
\mforall{}x,y:\mBbbR{}. (rpositive(x) {}\mRightarrow{} rpositive(y) {}\mRightarrow{} rpositive(x * y))
Date html generated:
2017_10_03-AM-08_23_29
Last ObjectModification:
2017_07_28-AM-07_22_53
Theory : reals
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