Nuprl Lemma : rpositive2_functionality
∀x,y:ℕ+ ⟶ ℤ. (bdd-diff(x;y) ⇒ (rpositive2(x) ⇐⇒ rpositive2(y)))
Proof
Definitions occuring in Statement :
rpositive2: rpositive2(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,
rpositive2: rpositive2(x),
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
guard: {T},
bdd-diff: bdd-diff(f;g),
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than: a < b,
squash: ↓T,
sq_type: SQType(T),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
multiply_functionality_wrt_le,
le_weakening,
le_functionality,
add-swap,
mul-commutes,
mul-associates,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
int_term_value_minus_lemma,
int_formula_prop_less_lemma,
itermMinus_wf,
intformless_wf,
minus-is-int-iff,
not_wf,
bnot_wf,
assert_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract-is-int-iff,
add-is-int-iff,
int_subtype_base,
multiply-is-int-iff,
lt_int_wf,
absval_ifthenelse,
one-mul,
mul-swap,
mul-distributes,
mul-distributes-right,
mul_bounds_1a,
subtract_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
nat_plus_properties,
nat_plus_subtype_nat,
mul_preserves_le,
all_wf,
le_wf,
less_than_wf,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
false_wf,
decidable__lt,
mul_nat_plus,
bdd-diff_inversion,
nat_plus_wf,
bdd-diff_wf,
rpositive2_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
lemma_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
functionEquality,
intEquality,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
because_Cache,
setElimination,
rename,
natural_numberEquality,
unionElimination,
voidElimination,
independent_isectElimination,
sqequalRule,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
minusEquality,
multiplyEquality,
int_eqEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
baseApply,
closedConclusion,
baseClosed,
pointwiseFunctionality,
promote_hyp,
imageElimination,
instantiate,
cumulativity,
impliesFunctionality
Latex:
\mforall{}x,y:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}. (bdd-diff(x;y) {}\mRightarrow{} (rpositive2(x) \mLeftarrow{}{}\mRightarrow{} rpositive2(y)))
Date html generated:
2016_05_18-AM-07_00_42
Last ObjectModification:
2016_01_17-AM-01_49_37
Theory : reals
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