Nuprl Lemma : rnonneg2_functionality
∀x,y:ℕ+ ⟶ ℤ. (bdd-diff(x;y) ⇒ (rnonneg2(x) ⇐⇒ rnonneg2(y)))
Proof
Definitions occuring in Statement :
rnonneg2: rnonneg2(x),
bdd-diff: bdd-diff(f;g),
nat_plus: ℕ+,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
bdd-diff: bdd-diff(f;g),
exists: ∃x:A. B[x],
rnonneg2: rnonneg2(x),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
guard: {T},
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
nat: ℕ,
so_lambda: λ2x.t[x],
int_upper: {i...},
le: A ≤ B,
uimplies: b supposing a,
so_apply: x[s],
subtype_rel: A ⊆r B,
sq_type: SQType(T),
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
cand: A c∧ B,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
nat_plus_wf,
rnonneg2_wf,
bdd-diff_wf,
bdd-diff_inversion,
mul_nat_plus,
less_than_wf,
int_upper_wf,
imax_wf,
all_wf,
le_wf,
less_than_transitivity1,
subtype_base_sq,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
imax_com,
iff_weakening_equal,
imax_nat_plus,
nat_plus_properties,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_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_formula_prop_eq_lemma,
int_formula_prop_wf,
imax_ub,
int_upper_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
itermMultiply_wf,
int_term_value_mul_lemma,
int_upper_subtype_int_upper,
absval_ifthenelse,
subtract_wf,
lt_int_wf,
subtract-is-int-iff,
multiply-is-int-iff,
itermSubtract_wf,
int_term_value_subtract_lemma,
false_wf,
assert_wf,
bnot_wf,
not_wf,
minus-is-int-iff,
itermMinus_wf,
int_term_value_minus_lemma,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
mul_preserves_le,
nat_plus_subtype_nat,
le_functionality,
le_weakening
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
functionExtensionality,
applyEquality,
hypothesisEquality,
functionEquality,
intEquality,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
setElimination,
rename,
multiplyEquality,
because_Cache,
lambdaEquality,
minusEquality,
independent_isectElimination,
instantiate,
cumulativity,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
applyLambdaEquality,
unionElimination,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
inlFormation,
inrFormation,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
impliesFunctionality
Latex:
\mforall{}x,y:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}. (bdd-diff(x;y) {}\mRightarrow{} (rnonneg2(x) \mLeftarrow{}{}\mRightarrow{} rnonneg2(y)))
Date html generated:
2017_10_03-AM-08_23_36
Last ObjectModification:
2017_07_28-AM-07_22_57
Theory : reals
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