Nuprl Lemma : bag-member-two-factorizations
∀[n:ℕ]. ∀[a,b:ℤ]. uiff(<a, b> ↓∈ two-factorizations(n);(1 ≤ a) ∧ (a ≤ n) ∧ ((a * b) = n ∈ ℤ))
Proof
Definitions occuring in Statement :
two-factorizations: two-factorizations(n),
bag-member: x ↓∈ bs,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
pair: <a, b>,
product: x:A × B[x],
multiply: n * m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
pi2: snd(t),
pi1: fst(t),
nat: ℕ,
prop: ℙ,
iff: P ⇐⇒ Q,
le: A ≤ B,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
bag-member: x ↓∈ bs,
squash: ↓T,
two-factorizations: two-factorizations(n),
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
guard: {T},
less_than: a < b,
cand: A c∧ B,
int_nzero: ℤ-o,
sq_stable: SqStable(P),
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
rev_uimplies: rev_uimplies(P;Q),
divides: b | a,
nat_plus: ℕ+,
less_than': less_than'(a;b),
true: True,
gt: i > j,
div_nrel: Div(a;n;q),
lelt: i ≤ j < k
Lemmas referenced :
bag-member-list,
decidable__equal_product,
decidable__equal_int,
two-factorizations_wf,
subtype_rel_list,
equal_wf,
less_than'_wf,
bag-member_wf,
list-subtype-bag,
le_wf,
equal-wf-base-T,
uiff_wf,
l_member_wf,
int_subtype_base,
nat_wf,
member-mapfilter,
less_than_wf,
from-upto_wf,
set_wf,
eq_int_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
equal-wf-base,
equal-wf-T-base,
assert_wf,
mapfilter_wf,
int_nzero_wf,
subtype_rel_sets,
nequal_wf,
int_nzero_properties,
intformnot_wf,
int_formula_prop_not_lemma,
exists_wf,
sq_stable__le,
decidable__le,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
assert_of_eq_int,
subtype_base_sq,
div_rem_sum,
itermMultiply_wf,
int_term_value_mul_lemma,
decidable__lt,
member-from-upto,
divides_iff_rem_zero,
div_unique2,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
pos_mul_arg_bounds,
intformimplies_wf,
intformor_wf,
int_formual_prop_imp_lemma,
int_formula_prop_or_lemma
Rules used in proof :
cut,
addLevel,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
independent_pairFormation,
isect_memberFormation,
introduction,
independent_isectElimination,
extract_by_obid,
isectElimination,
productEquality,
intEquality,
independent_functionElimination,
lambdaFormation,
because_Cache,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
independent_pairEquality,
applyEquality,
setEquality,
multiplyEquality,
setElimination,
rename,
voidElimination,
cumulativity,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
baseApply,
closedConclusion,
isect_memberEquality,
addEquality,
dependent_set_memberEquality,
remainderEquality,
dependent_pairFormation,
int_eqEquality,
voidEquality,
computeAll,
divideEquality,
applyLambdaEquality,
unionElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a,b:\mBbbZ{}]. uiff(<a, b> \mdownarrow{}\mmember{} two-factorizations(n);(1 \mleq{} a) \mwedge{} (a \mleq{} n) \mwedge{} ((a * b) = n))
Date html generated:
2018_05_21-PM-09_06_14
Last ObjectModification:
2017_07_26-PM-06_28_59
Theory : general
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