Nuprl Lemma : neg_mul_arg_bounds
∀a,b:ℤ. (a * b < 0 ⇐⇒ (a < 0 ∧ (b > 0)) ∨ ((a > 0) ∧ b < 0))
Proof
Definitions occuring in Statement :
less_than: a < b,
gt: i > j,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
label: ...$L... t,
gt: i > j,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
uiff: uiff(P;Q)
Lemmas referenced :
pos_mul_arg_bounds,
istype-int,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermMinus_wf,
itermVar_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
gt_wf,
less_than_wf,
itermConstant_wf,
int_term_value_constant_lemma,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
iff_weakening_uiff,
minus_mono_wrt_lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
minusEquality,
hypothesisEquality,
Error :inhabitedIsType,
hypothesis,
productElimination,
because_Cache,
unionElimination,
isectElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
Error :universeIsType,
multiplyEquality,
Error :unionIsType,
Error :productIsType,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
independent_pairFormation,
promote_hyp,
Error :inlFormation_alt,
Error :inrFormation_alt
Latex:
\mforall{}a,b:\mBbbZ{}. (a * b < 0 \mLeftarrow{}{}\mRightarrow{} (a < 0 \mwedge{} (b > 0)) \mvee{} ((a > 0) \mwedge{} b < 0))
Date html generated:
2019_06_20-PM-01_13_34
Last ObjectModification:
2018_10_03-PM-11_00_35
Theory : int_2
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