Nuprl Lemma : rnonneg-iff
∀[x:ℝ]. (rnonneg(x) ⇐⇒ rnonneg2(x))
Proof
Definitions occuring in Statement :
rnonneg2: rnonneg2(x),
rnonneg: rnonneg(x),
real: ℝ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
rnonneg2: rnonneg2(x),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
real: ℝ,
rev_implies: P ⇐ Q,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
int_upper: {i...},
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
sq_stable: SqStable(P),
regular-int-seq: k-regular-seq(f),
nat: ℕ,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
squash: ↓T,
pi1: fst(t),
subtract: n - m,
less_than: a < b,
true: True
Lemmas referenced :
mul_preserves_lt,
imax_strict_ub,
mul_nat_plus,
imax_ub,
imax_wf,
add-swap,
add-commutes,
mul-commutes,
mul-swap,
minus-one-mul,
minus-add,
mul-associates,
mul-distributes,
equal_wf,
int_term_value_add_lemma,
itermAdd_wf,
subtract-is-int-iff,
add-is-int-iff,
multiply-is-int-iff,
int_subtype_base,
minus-is-int-iff,
int_upper_subtype_nat,
add_nat_wf,
false_wf,
mul_bounds_1a,
subtract_wf,
absval_ubound,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
subtype_rel_sets,
sq_stable__le,
le_weakening,
le_functionality,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
int_upper_properties,
nat_plus_subtype_nat,
mul_preserves_le,
real_wf,
less_than_wf,
less_than_transitivity1,
le_wf,
int_upper_wf,
exists_wf,
all_wf,
less_than'_wf,
rnonneg_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
cut,
lemma_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
introduction,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
voidElimination,
applyEquality,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
multiplyEquality,
dependent_set_memberEquality,
independent_isectElimination,
dependent_pairFormation,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
promote_hyp,
addEquality,
independent_functionElimination,
setEquality,
imageMemberEquality,
baseClosed,
imageElimination,
baseApply,
closedConclusion,
pointwiseFunctionality,
inrFormation,
inlFormation
Latex:
\mforall{}[x:\mBbbR{}]. (rnonneg(x) \mLeftarrow{}{}\mRightarrow{} rnonneg2(x))
Date html generated:
2016_05_18-AM-07_01_40
Last ObjectModification:
2016_01_17-AM-01_49_41
Theory : reals
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