Nuprl Lemma : unique_mfact
∀g:IAbMonoid
(Cancel(|g|;|g|;*) ⇒ (∀a,b:|g|. Dec(a | b)) ⇒ (∀ps,qs:Prime(g) List. (((Π ps) ~ (Π qs)) ⇒ ps ≡ qs upto ~)))
Proof
Definitions occuring in Statement :
permr_massoc: as ≡ bs upto ~,
mprime_ty: Prime(g),
massoc: a ~ b,
mdivides: b | a,
mon_reduce: mon_reduce,
list: T List,
decidable: Dec(P),
all: ∀x:A. B[x],
implies: P ⇒ Q,
iabmonoid: IAbMonoid,
grp_op: *,
grp_car: |g|,
cancel: Cancel(T;S;op)
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
iabmonoid: IAbMonoid,
imon: IMonoid,
so_lambda: λ2x.t[x],
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
mprime_ty: Prime(g),
so_apply: x[s],
mon_reduce: mon_reduce,
top: Top,
infix_ap: x f y,
massoc: a ~ b,
symmetrize: Symmetrize(x,y.R[x; y];a;b),
and: P ∧ Q,
munit: g-unit(u),
or: P ∨ Q,
cons: [a / b],
false: False,
mprime: IsPrime(a),
not: ¬A,
mdivides: b | a,
exists: ∃x:A. B[x],
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
sq_stable: SqStable(P),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than: a < b,
stable: Stable{P},
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
mprime_ty_wf,
all_wf,
list_wf,
massoc_wf,
mon_reduce_wf,
subtype_rel_list,
grp_car_wf,
permr_massoc_wf,
reduce_nil_lemma,
istype-void,
grp_id_wf,
reduce_cons_lemma,
grp_op_wf,
decidable_wf,
mdivides_wf,
cancel_wf,
iabmonoid_wf,
list-cases,
product_subtype_list,
permr_massoc_weakening,
nil_wf,
permr_weakening,
munit_of_op,
equal_wf,
squash_wf,
true_wf,
istype-universe,
mon_assoc,
subtype_rel_self,
iff_weakening_equal,
mprime_divs_list_el,
sq_stable__mprime,
mdivisor_of_atom_is_assoc2,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
mprime_imp_matomic,
not_wf,
munit_wf,
permr_massoc_functionality,
cons_wf,
reject_wf,
permr_inversion,
select_reject_permr,
grp_sig_wf,
mon_reduce_functionality_wrt_permr,
massoc_functionality_wrt_massoc,
grp_op_functionality_wrt_massoc,
massoc_weakening,
massoc_cancel,
cons_functionality_wrt_permr_massoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
dependent_functionElimination,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality_alt,
functionEquality,
hypothesisEquality,
applyEquality,
independent_isectElimination,
universeIsType,
inhabitedIsType,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
functionIsType,
productElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
dependent_pairFormation_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
equalityIsType1,
approximateComputation,
int_eqEquality,
independent_pairFormation,
isect_memberFormation_alt,
functionIsTypeImplies,
applyLambdaEquality
Latex:
\mforall{}g:IAbMonoid
(Cancel(|g|;|g|;*)
{}\mRightarrow{} (\mforall{}a,b:|g|. Dec(a | b))
{}\mRightarrow{} (\mforall{}ps,qs:Prime(g) List. (((\mPi{} ps) \msim{} (\mPi{} qs)) {}\mRightarrow{} ps \mequiv{} qs upto \msim{})))
Date html generated:
2019_10_16-PM-01_05_58
Last ObjectModification:
2018_10_08-PM-00_12_50
Theory : factor_1
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