Nuprl Lemma : trial-division_wf
∀[n:ℕ+]. ∀[L:{2...} List]. (trial-division(n;L) ∈ ∃n1:ℤ [(n1 < n ∧ (2 ≤ n1) ∧ (n1 | n))] + Top)
Proof
Definitions occuring in Statement :
trial-division: trial-division(n;L),
divides: b | a,
list: T List,
int_upper: {i...},
nat_plus: ℕ+,
less_than: a < b,
uall: ∀[x:A]. B[x],
top: Top,
le: A ≤ B,
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
member: t ∈ T,
union: left + right,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
trial-division: trial-division(n;L),
list_ind: list_ind,
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
int_upper: {i...},
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
has-value: (a)↓,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
cand: A c∧ B,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
int_upper_wf,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
sq_exists_wf,
le_wf,
divides_wf,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
list_ind_cons_lemma,
top_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
value-type-has-value,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
list_wf,
nat_plus_wf,
gcd_wf,
int_upper_properties,
nat_plus_properties,
gcd_is_divisor_2,
better-gcd-gcd,
divisor_bound,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
gcd_is_divisor_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
because_Cache,
unionElimination,
inrEquality,
productEquality,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
cumulativity,
imageElimination,
unionEquality,
equalityElimination,
callbyvalueReduce,
inlEquality
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[L:\{2...\} List]. (trial-division(n;L) \mmember{} \mexists{}n1:\mBbbZ{} [(n1 < n \mwedge{} (2 \mleq{} n1) \mwedge{} (n1 | n))] + Top)
Date html generated:
2018_05_21-PM-08_15_50
Last ObjectModification:
2017_07_26-PM-05_50_06
Theory : general
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