Nuprl Lemma : bag-to-list_wf
∀[T:Type]
∀[b:bag(T)]. ∀[cmp:comparison({x:T| x ↓∈ b} )].
bag-to-list(cmp;b) ∈ T List supposing ∀x,y:{x:T| x ↓∈ b} . (((cmp x y) = 0 ∈ ℤ) ⇒ (x = y ∈ T))
supposing valueall-type(T)
Proof
Definitions occuring in Statement :
bag-to-list: bag-to-list(cmp;b),
bag-member: x ↓∈ bs,
bag: bag(T),
comparison: comparison(T),
list: T List,
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
squash: ↓T,
prop: ℙ,
linorder: Linorder(T;x,y.R[x; y]),
order: Order(T;x,y.R[x; y]),
cand: A c∧ B,
comparison: comparison(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bag-to-list: bag-to-list(cmp;b),
le: A ≤ B,
sorted-by: sorted-by(R;L),
int_seg: {i..j-},
sq_stable: SqStable(P),
not: ¬A,
false: False
Lemmas referenced :
list_wf,
member-permutation,
bag-member-iff-sq-list-member,
l_member_wf,
bag-member_wf,
list-subtype-bag,
comparison-refl,
comparison-trans,
comparison-antisym,
equal-wf-T-base,
set_wf,
comparison-connex,
subtype_rel_comparison,
ext-eq_inversion,
subtype_rel_weakening,
equal-wf-base,
permutation_wf,
all_wf,
equal_wf,
comparison_wf,
bag_wf,
valueall-type_wf,
list-subtype,
subtype_rel_list,
permutation-sorted-by-unique,
le_wf,
comparison-sort_wf,
set-valueall-type,
sorted-by_wf,
sq_stable__all,
int_seg_wf,
length_wf,
sq_stable__le,
less_than'_wf,
squash_wf,
comparison-sort-permutation,
permutation_functionality_wrt_permutation,
permutation-strong-subtype,
strong-subtype-set2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
independent_pairFormation,
lambdaEquality,
setElimination,
rename,
independent_isectElimination,
imageElimination,
dependent_set_memberEquality,
setEquality,
applyEquality,
imageMemberEquality,
baseClosed,
lambdaFormation,
intEquality,
promote_hyp,
productEquality,
axiomEquality,
functionEquality,
isect_memberEquality,
universeEquality,
natural_numberEquality,
independent_pairEquality,
voidElimination
Latex:
\mforall{}[T:Type]
\mforall{}[b:bag(T)]. \mforall{}[cmp:comparison(\{x:T| x \mdownarrow{}\mmember{} b\} )].
bag-to-list(cmp;b) \mmember{} T List supposing \mforall{}x,y:\{x:T| x \mdownarrow{}\mmember{} b\} . (((cmp x y) = 0) {}\mRightarrow{} (x = y))
supposing valueall-type(T)
Date html generated:
2017_10_01-AM-08_56_53
Last ObjectModification:
2017_07_26-PM-04_39_08
Theory : bags
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