Nuprl Lemma : bag-remove1_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[bs:bag(T)]. (bag-remove1(eq;bs;x) ∈ bag(T)?)
Proof
Definitions occuring in Statement :
bag-remove1: bag-remove1(eq;bs;a),
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
unit: Unit,
member: t ∈ T,
union: left + right,
universe: Type
Definitions unfolded in proof :
bag: bag(T),
member: t ∈ T,
uall: ∀[x:A]. B[x],
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
guard: {T},
or: P ∨ Q,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
bag-append: as + bs,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
true: True,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
rev_implies: P ⇐ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
not: ¬A,
false: False
Lemmas referenced :
bag_wf,
unit_wf2,
list_wf,
permutation_wf,
equal_wf,
equal-wf-base,
deq_wf,
member-permutation,
bag-remove1-property1,
subtype_rel_union,
list-subtype-bag,
quotient-member-eq,
permutation-equiv,
append_wf,
squash_wf,
true_wf,
bag-append_wf,
reverse-bag,
iff_weakening_equal,
permutation_transitivity,
permutation_weakening,
cons_wf,
nil_wf,
permutation_functionality_wrt_permutation,
append_functionality_wrt_permutation,
permutation-rotate,
list_ind_cons_lemma,
list_ind_nil_lemma,
cons_cancel_wrt_permutation,
reverse_wf,
length_wf_nat,
nat_wf,
member_append,
cons_member,
l_member_wf,
bag-remove1_wf1
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
pointwiseFunctionalityForEquality,
unionEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
because_Cache,
rename,
dependent_functionElimination,
independent_functionElimination,
productEquality,
universeEquality,
isect_memberFormation,
axiomEquality,
isect_memberEquality,
unionElimination,
hyp_replacement,
applyLambdaEquality,
applyEquality,
independent_isectElimination,
lambdaEquality,
natural_numberEquality,
inlEquality,
imageElimination,
imageMemberEquality,
baseClosed,
voidElimination,
voidEquality,
dependent_set_memberEquality,
inlFormation,
inrFormation,
setElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[bs:bag(T)]. (bag-remove1(eq;bs;x) \mmember{} bag(T)?)
Date html generated:
2018_05_21-PM-09_48_11
Last ObjectModification:
2017_07_26-PM-06_30_34
Theory : bags_2
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