Nuprl Lemma : bag-admissable-well-founded
∀k:ℕ
∀[T:Type]
(T ~ ℕk
⇒ (∀[R:bag(T) ⟶ bag(T) ⟶ ℙ]
((∀as,bs:bag(T). Dec(R[as;bs]))
⇒ Order(bag(T);as,bs.R[as;bs])
⇒ bag-admissable(T;as,bs.R[as;bs])
⇒ WellFnd{i}(bag(T);as,bs.R[as;bs] ∧ (¬(as = bs ∈ bag(T)))))))
Proof
Definitions occuring in Statement :
bag-admissable: bag-admissable(T;as,bs.R[as; bs]),
bag: bag(T),
equipollent: A ~ B,
order: Order(T;x,y.R[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,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
not: ¬A,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
trans: Trans(T;x,y.E[x; y]),
cand: A c∧ B,
guard: {T},
order: Order(T;x,y.R[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y])
Lemmas referenced :
bag-ordering-wellfounded,
bag_wf,
not_wf,
equal_wf,
sub-bag_wf,
bag-admissable_wf,
order_wf,
all_wf,
decidable_wf,
equipollent_wf,
int_seg_wf,
nat_wf,
and_wf,
decidable__equal_equipollent,
decidable__cand,
decidable__not,
decidable__equal_bag,
sub-bag-admissable
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
independent_functionElimination,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
functionExtensionality,
cumulativity,
universeEquality,
voidElimination,
because_Cache,
functionEquality,
natural_numberEquality,
setElimination,
rename,
productElimination,
independent_pairFormation,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
setEquality,
isect_memberEquality
Latex:
\mforall{}k:\mBbbN{}
\mforall{}[T:Type]
(T \msim{} \mBbbN{}k
{}\mRightarrow{} (\mforall{}[R:bag(T) {}\mrightarrow{} bag(T) {}\mrightarrow{} \mBbbP{}]
((\mforall{}as,bs:bag(T). Dec(R[as;bs]))
{}\mRightarrow{} Order(bag(T);as,bs.R[as;bs])
{}\mRightarrow{} bag-admissable(T;as,bs.R[as;bs])
{}\mRightarrow{} WellFnd\{i\}(bag(T);as,bs.R[as;bs] \mwedge{} (\mneg{}(as = bs))))))
Date html generated:
2016_10_25-AM-11_31_28
Last ObjectModification:
2016_07_12-AM-07_36_11
Theory : bags_2
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