Nuprl Lemma : atomic_char
∀a:ℤ. (atomic(a) ⇐⇒ {(¬(a ~ 1)) ∧ (∀b:ℤ. ((b | a) ⇒ ((b ~ 1) ∨ (b ~ a))))})
Proof
Definitions occuring in Statement :
atomic: atomic(a),
assoced: a ~ b,
divides: b | a,
guard: {T},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
or: P ∨ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
false: False,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
atomic: atomic(a),
guard: {T},
uiff: uiff(P;Q),
uimplies: b supposing a,
stable: Stable{P},
reducible: reducible(a),
exists: ∃x:A. B[x],
divides: b | a,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
cand: A c∧ B,
top: Top,
true: True,
sq_type: SQType(T),
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P)
Lemmas referenced :
or_wf,
all_wf,
reducible_wf,
int_subtype_base,
equal-wf-base,
not_wf,
divides_wf,
assoced_wf,
not_over_or,
decidable__assoced,
decidable__or,
stable__from_decidable,
zero_ann_b,
nequal_wf,
istype-void,
set_subtype_base,
assoced_functionality_wrt_assoced,
assoced_weakening,
multiply_functionality_wrt_assoced,
mul-commutes,
one-mul,
any_divs_zero,
subtype_base_sq,
assoced_elim,
equal-wf-base-T,
int_formula_prop_wf,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermMultiply_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
int_nzero_properties,
int_term_value_constant_lemma,
itermConstant_wf,
mul_cancel_in_assoced,
int_entire,
assoced_inversion
Rules used in proof :
functionEquality,
lambdaEquality,
because_Cache,
baseClosed,
applyEquality,
productEquality,
intEquality,
natural_numberEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
voidElimination,
independent_functionElimination,
hypothesis,
productElimination,
sqequalHypSubstitution,
thin,
cut,
independent_pairFormation,
lambdaFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution,
independent_isectElimination,
dependent_functionElimination,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
Error :dependent_pairFormation_alt,
Error :dependent_set_memberEquality_alt,
Error :universeIsType,
Error :lambdaFormation_alt,
Error :productIsType,
Error :inhabitedIsType,
Error :functionIsType,
setElimination,
rename,
Error :equalityIstype,
baseApply,
closedConclusion,
Error :lambdaEquality_alt,
sqequalBase,
equalityTransitivity,
multiplyEquality,
Error :isect_memberEquality_alt,
cumulativity,
instantiate,
unionElimination,
promote_hyp,
orFunctionality,
addLevel,
dependent_pairFormation,
computeAll,
voidEquality,
isect_memberEquality,
int_eqEquality,
dependent_set_memberEquality
Latex:
\mforall{}a:\mBbbZ{}. (atomic(a) \mLeftarrow{}{}\mRightarrow{} \{(\mneg{}(a \msim{} 1)) \mwedge{} (\mforall{}b:\mBbbZ{}. ((b | a) {}\mRightarrow{} ((b \msim{} 1) \mvee{} (b \msim{} a))))\})
Date html generated:
2019_06_20-PM-02_22_51
Last ObjectModification:
2019_01_13-AM-09_09_05
Theory : num_thy_1
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