Nuprl Lemma : bag-remove1-append1
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x,y:T]. ∀[bs:bag(T)].
(bag-remove1(eq;{y} + bs;x)
= if eq x y then inl bs else case bag-remove1(eq;bs;x) of inl(as) => inl ({y} + as) | inr(x) => inr ⋅ fi
∈ (bag(T)?))
Proof
Definitions occuring in Statement :
bag-remove1: bag-remove1(eq;bs;a),
bag-append: as + bs,
single-bag: {x},
bag: bag(T),
deq: EqDecider(T),
ifthenelse: if b then t else f fi ,
it: ⋅,
uall: ∀[x:A]. B[x],
unit: Unit,
apply: f a,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
inr: inr x ,
inl: inl x,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
deq: EqDecider(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
prop: ℙ,
isl: isl(x),
sq_or: a ↓∨ b
Lemmas referenced :
eqtt_to_assert,
safe-assert-deq,
equal_wf,
bag_wf,
unit_wf2,
bag-remove1_wf,
bag-append_wf,
single-bag_wf,
iff_weakening_equal,
bag-remove1-member,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
bag-remove1-property,
squash_wf,
true_wf,
istype-universe,
inl-one-one,
not-0-eq-1,
inr-one-one,
it_wf,
subtype_rel_self,
btrue_wf,
bfalse_wf,
btrue_neq_bfalse,
bag-append-assoc-comm,
bag-append-cancel,
bag-member-single,
bag-member_wf,
bag-member-append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
extract_by_obid,
isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
lambdaEquality_alt,
imageElimination,
because_Cache,
universeIsType,
universeEquality,
unionEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
inlEquality_alt,
independent_functionElimination,
dependent_pairFormation_alt,
equalityIsType3,
promote_hyp,
instantiate,
voidElimination,
equalityIsType1,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
applyLambdaEquality,
inrEquality_alt,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
inrFormation_alt,
inlFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x,y:T]. \mforall{}[bs:bag(T)].
(bag-remove1(eq;\{y\} + bs;x)
= if eq x y
then inl bs
else case bag-remove1(eq;bs;x) of inl(as) => inl (\{y\} + as) | inr(x) => inr \mcdot{}
fi )
Date html generated:
2019_10_16-AM-11_31_08
Last ObjectModification:
2018_10_16-AM-09_34_29
Theory : bags_2
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