Nuprl Lemma : mfact_exists
∀g:IAbMonoid
(Cancel(|g|;|g|;*)
⇒ WellFnd{i}(|g|;x,y.x p| y)
⇒ (∀c:|g|. Dec(Reducible(c)))
⇒ (∀b:|g|. ((¬(g-unit(b))) ⇒ (∃as:Atom{g} List. (b = (Π as) ∈ |g|)))))
Proof
Definitions occuring in Statement :
matom_ty: Atom{g},
mreducible: Reducible(a),
mpdivides: a p| b,
munit: g-unit(u),
mon_reduce: mon_reduce,
list: T List,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
decidable: Dec(P),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
equal: s = t ∈ T,
iabmonoid: IAbMonoid,
grp_op: *,
grp_car: |g|,
cancel: Cancel(T;S;op)
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
iabmonoid: IAbMonoid,
imon: IMonoid,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
matom_ty: Atom{g},
so_apply: x[s],
exists: ∃x:A. B[x],
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
mreducible: Reducible(a),
and: P ∧ Q,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y,
mon_reduce: mon_reduce,
top: Top,
matomic: Atomic(a),
cand: A c∧ B
Lemmas referenced :
not_wf,
munit_wf,
grp_car_wf,
decidable_wf,
mreducible_wf,
wellfounded_wf,
mpdivides_wf,
cancel_wf,
grp_op_wf,
iabmonoid_wf,
exists_wf,
list_wf,
matom_ty_wf,
equal_wf,
mon_reduce_wf,
subtype_rel_list,
non_munit_diff_imp_mpdivides,
squash_wf,
true_wf,
istype-universe,
abmonoid_comm,
subtype_rel_self,
iff_weakening_equal,
append_wf,
mon_reduce_append,
cons_wf,
nil_wf,
reduce_cons_lemma,
istype-void,
reduce_nil_lemma,
matomic_wf,
mon_ident
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
functionIsType,
because_Cache,
lambdaEquality_alt,
inhabitedIsType,
functionEquality,
applyEquality,
independent_isectElimination,
independent_functionElimination,
productIsType,
equalityIsType1,
unionElimination,
productElimination,
equalitySymmetry,
imageElimination,
equalityTransitivity,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
dependent_pairFormation_alt,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
independent_pairFormation
Latex:
\mforall{}g:IAbMonoid
(Cancel(|g|;|g|;*)
{}\mRightarrow{} WellFnd\{i\}(|g|;x,y.x p| y)
{}\mRightarrow{} (\mforall{}c:|g|. Dec(Reducible(c)))
{}\mRightarrow{} (\mforall{}b:|g|. ((\mneg{}(g-unit(b))) {}\mRightarrow{} (\mexists{}as:Atom\{g\} List. (b = (\mPi{} as))))))
Date html generated:
2019_10_16-PM-01_05_52
Last ObjectModification:
2018_10_08-PM-00_15_50
Theory : factor_1
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