Nuprl Lemma : posint_atom_imp_prime
∀a:Atom{<ℤ+,*>}. IsPrime(a)
Proof
Definitions occuring in Statement :
posint_mul_mon: <ℤ+,*>,
matom_ty: Atom{g},
mprime: IsPrime(a),
all: ∀x:A. B[x]
Definitions unfolded in proof :
mprime: IsPrime(a),
matom_ty: Atom{g},
posint_mul_mon: <ℤ+,*>,
grp_car: |g|,
pi1: fst(t),
grp_op: *,
pi2: snd(t),
infix_ap: x f y,
munit: g-unit(u),
matomic: Atomic(a),
grp_id: e,
mreducible: Reducible(a),
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
member: t ∈ T,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
exists: ∃x:A. B[x],
divides: b | a,
mdivides: b | a,
iff: P ⇐⇒ Q,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rev_implies: P ⇐ Q,
gt: i > j,
prime: prime(a),
atomic: atomic(a),
reducible: reducible(a),
int_nzero: ℤ-o,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
nequal: a ≠ b ∈ T ,
le: A ≤ B,
subtract: n - m,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nat: ℕ,
sq_stable: SqStable(P),
assoced: a ~ b
Lemmas referenced :
mdivides_wf,
posint_mul_mon_wf,
less_than_wf,
mul_nat_plus,
nat_plus_wf,
not_wf,
exists_wf,
equal-wf-base,
set_subtype_base,
istype-int,
int_subtype_base,
nat_plus_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermMultiply_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
pos_mul_arg_bounds,
decidable__lt,
intformless_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
divides_wf,
mon_subtype_grp_sig,
abmonoid_subtype_mon,
subtype_rel_transitivity,
abmonoid_wf,
mon_wf,
grp_sig_wf,
atomic_imp_prime,
unit_chars,
assoced_wf,
reducible_wf,
absval_unfold,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
istype-false,
not-lt-2,
not-equal-2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-swap,
add-commutes,
zero-add,
le-add-cancel,
condition-implies-le,
minus-add,
minus-zero,
add-zero,
eqff_to_assert,
nequal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
int_nzero_properties,
itermMinus_wf,
int_term_value_minus_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
absval_mul,
subtype_rel_self,
iff_weakening_equal,
absval_pos,
sq_stable__and,
sq_stable__not,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
le_wf,
one_divs_any,
absval_wf,
grp_car_wf,
assoced_functionality_wrt_assoced,
absval_assoced,
assoced_weakening
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation_alt,
cut,
setElimination,
thin,
rename,
sqequalHypSubstitution,
productElimination,
hypothesis,
independent_functionElimination,
voidElimination,
universeIsType,
introduction,
extract_by_obid,
dependent_functionElimination,
applyEquality,
because_Cache,
hypothesisEquality,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
isectElimination,
inhabitedIsType,
setIsType,
productIsType,
lambdaEquality_alt,
productEquality,
baseApply,
closedConclusion,
intEquality,
independent_isectElimination,
equalityIsType4,
dependent_pairFormation_alt,
applyLambdaEquality,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
multiplyEquality,
equalityTransitivity,
equalitySymmetry,
inlFormation_alt,
instantiate,
inrFormation_alt,
minusEquality,
equalityElimination,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
imageElimination,
addEquality,
equalityIsType2,
promote_hyp,
cumulativity,
equalityIsType1,
universeEquality,
functionIsTypeImplies
Latex:
\mforall{}a:Atom\{<\mBbbZ{}\msupplus{},*>\}. IsPrime(a)
Date html generated:
2019_10_16-PM-01_06_16
Last ObjectModification:
2018_10_08-PM-05_38_49
Theory : factor_1
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