Nuprl Lemma : bag-ordering-wellfounded
∀p:ℕ
∀[T:Type]
(T ~ ℕp
⇒ (∀[R:bag(T) ⟶ bag(T) ⟶ ℙ]
(Trans(bag(T);a,b.R[a;b])
⇒ (∀a,b:bag(T). Dec(R[a;b]))
⇒ (∀a,b:bag(T). (sub-bag(T;a;b) ⇒ (¬R[b;a])))
⇒ WellFnd{i}(bag(T);a,b.R[a;b]))))
Proof
Definitions occuring in Statement :
sub-bag: sub-bag(T;as;bs),
bag: bag(T),
equipollent: A ~ B,
trans: Trans(T;x,y.E[x; y]),
int_seg: {i..j-},
nat: ℕ,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
nat: ℕ,
trans: Trans(T;x,y.E[x; y]),
no-descending-chain: no-descending-chain(T;<),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A
Lemmas referenced :
no-descending-chain-implies-wellfounded,
bag_wf,
all_wf,
sub-bag_wf,
not_wf,
decidable_wf,
trans_wf,
equipollent_wf,
int_seg_wf,
nat_wf,
bag-dickson-lemma,
int_seg_subtype_nat,
false_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality,
natural_numberEquality,
setElimination,
rename,
dependent_functionElimination,
productElimination,
dependent_pairFormation,
independent_isectElimination,
independent_pairFormation
Latex:
\mforall{}p:\mBbbN{}
\mforall{}[T:Type]
(T \msim{} \mBbbN{}p
{}\mRightarrow{} (\mforall{}[R:bag(T) {}\mrightarrow{} bag(T) {}\mrightarrow{} \mBbbP{}]
(Trans(bag(T);a,b.R[a;b])
{}\mRightarrow{} (\mforall{}a,b:bag(T). Dec(R[a;b]))
{}\mRightarrow{} (\mforall{}a,b:bag(T). (sub-bag(T;a;b) {}\mRightarrow{} (\mneg{}R[b;a])))
{}\mRightarrow{} WellFnd\{i\}(bag(T);a,b.R[a;b]))))
Date html generated:
2016_05_15-PM-08_10_03
Last ObjectModification:
2015_12_27-PM-04_12_25
Theory : bags_2
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