Nuprl Lemma : rpositive-iff
∀[x:ℝ]. (rpositive(x) ⇐⇒ rpositive2(x))
Proof
Definitions occuring in Statement :
rpositive2: rpositive2(x),
rpositive: rpositive(x),
real: ℝ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
rpositive2: rpositive2(x),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
real: ℝ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
rpositive: rpositive(x),
sq_exists: ∃x:{A| B[x]},
all: ∀x:A. B[x],
uimplies: b supposing a,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
regular-int-seq: k-regular-seq(f),
subtype_rel: A ⊆r B,
nat: ℕ,
true: True,
less_than': less_than'(a;b)
Lemmas referenced :
rpositive_wf,
exists_wf,
nat_plus_wf,
all_wf,
le_wf,
real_wf,
rnonzero-lemma1,
absval_ifthenelse,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
nat_plus_properties,
decidable__le,
less_than_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than'_wf,
assert_wf,
bnot_wf,
not_wf,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
mul_preserves_le,
nat_plus_subtype_nat,
multiply-is-int-iff,
int_subtype_base,
minus-is-int-iff,
false_wf,
mul_preserves_lt,
squash_wf,
true_wf,
absval_pos,
subtract_wf,
decidable__lt,
itermSubtract_wf,
itermMultiply_wf,
int_term_value_subtract_lemma,
int_term_value_mul_lemma,
mul_nat_plus,
itermAdd_wf,
int_term_value_add_lemma,
iff_weakening_equal,
mul_cancel_in_lt
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
lambdaEquality,
functionEquality,
because_Cache,
multiplyEquality,
applyEquality,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
dependent_set_memberEquality,
imageElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
independent_pairEquality,
axiomEquality,
impliesFunctionality,
functionExtensionality,
baseClosed,
baseApply,
closedConclusion,
pointwiseFunctionality,
addEquality,
imageMemberEquality,
universeEquality,
dependent_set_memberFormation
Latex:
\mforall{}[x:\mBbbR{}]. (rpositive(x) \mLeftarrow{}{}\mRightarrow{} rpositive2(x))
Date html generated:
2017_10_03-AM-08_23_14
Last ObjectModification:
2017_07_28-AM-07_22_46
Theory : reals
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