Nuprl Lemma : primality-test
∀b:ℕ. (prime(b)) supposing ((∀p:ℕ. (prime(p) ⇒ ((p * p) ≤ b) ⇒ (¬(p | b)))) and (2 ≤ b))
Proof
Definitions occuring in Statement :
prime: prime(a),
divides: b | a,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
atomic: atomic(a),
cand: A c∧ B,
iff: P ⇐⇒ Q,
reducible: reducible(a),
int_nzero: ℤ-o,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
divides: b | a,
true: True
Lemmas referenced :
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
int_seg_subtype,
false_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
le_wf,
less_than'_wf,
divides_wf,
prime_wf,
nat_wf,
all_wf,
not_wf,
isect_wf,
decidable__lt,
lelt_wf,
set_wf,
less_than_wf,
primrec-wf2,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
atomic_imp_prime,
equal-wf-T-base,
assoced_nelim,
assoced_wf,
reducible_wf,
absval_assoced,
assoced_inversion,
absval_wf,
assoced_transitivity,
subtype_base_sq,
int_subtype_base,
int_nzero_properties,
absval_ifthenelse,
equal_wf,
exists_wf,
lt_int_wf,
bool_wf,
assert_wf,
le_int_wf,
bnot_wf,
itermMinus_wf,
int_term_value_minus_lemma,
mul_preserves_lt,
itermMultiply_wf,
int_term_value_mul_lemma,
absval_nat_plus,
multiply-is-int-iff,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
decidable__equal_int_seg,
divides_transitivity,
int_seg_subtype_nat,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
mul_preserves_le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
because_Cache,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
unionElimination,
addLevel,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
isect_memberFormation,
independent_pairEquality,
axiomEquality,
multiplyEquality,
functionEquality,
addEquality,
baseClosed,
impliesFunctionality,
independent_functionElimination,
promote_hyp,
instantiate,
cumulativity,
minusEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
equalityElimination
Latex:
\mforall{}b:\mBbbN{}. (prime(b)) supposing ((\mforall{}p:\mBbbN{}. (prime(p) {}\mRightarrow{} ((p * p) \mleq{} b) {}\mRightarrow{} (\mneg{}(p | b)))) and (2 \mleq{} b))
Date html generated:
2018_05_21-PM-06_55_18
Last ObjectModification:
2017_07_26-PM-04_59_22
Theory : general
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