Nuprl Lemma : bag-remove1-equal
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[as,bs:bag(T)]. ∀[a,b:T].
uiff(bag-remove1(eq;as;a) = bag-remove1(eq;bs;b) ∈ (bag(T)?);(({a} + bs) = ({b} + as) ∈ bag(T))
↓∨ ((¬a ↓∈ as) ∧ (¬b ↓∈ bs)))
Proof
Definitions occuring in Statement :
bag-remove1: bag-remove1(eq;bs;a),
bag-member: x ↓∈ bs,
bag-append: as + bs,
single-bag: {x},
bag: bag(T),
deq: EqDecider(T),
sq_or: a ↓∨ b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
unit: Unit,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
sq_or: a ↓∨ b,
squash: ↓T,
all: ∀x:A. B[x],
prop: ℙ,
or: P ∨ Q,
exists: ∃x:A. B[x],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
isl: isl(x),
cand: A c∧ B,
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
outl: outl(x),
assert: ↑b,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
exposed-bfalse: exposed-bfalse
Lemmas referenced :
unit_wf2,
bag-remove1_wf,
sq_or_wf,
equal_wf,
bag_wf,
bag-append_wf,
single-bag_wf,
not_wf,
bag-member_wf,
istype-universe,
bag-remove1-property,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
istype-void,
bag-append-assoc-comm,
btrue_wf,
bfalse_wf,
btrue_neq_bfalse,
deq_wf,
bag-remove1-member,
bag-remove1-append1,
eqtt_to_assert,
safe-assert-deq,
outl_wf,
assert_wf,
isl_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
bag-remove1-non-member,
bag-member-append,
bag-member-single
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
imageElimination,
hypothesis,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
equalityIsType1,
unionIsType,
inhabitedIsType,
universeIsType,
extract_by_obid,
isectElimination,
dependent_functionElimination,
productEquality,
productElimination,
independent_pairEquality,
isect_memberEquality_alt,
because_Cache,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
universeEquality,
unionElimination,
inlFormation_alt,
applyEquality,
lambdaEquality_alt,
unionEquality,
natural_numberEquality,
instantiate,
independent_isectElimination,
independent_functionElimination,
applyLambdaEquality,
productIsType,
functionIsType,
dependent_set_memberEquality_alt,
equalityIsType3,
setElimination,
rename,
voidElimination,
inrFormation_alt,
lambdaFormation_alt,
equalityElimination,
promote_hyp,
hyp_replacement,
inlEquality_alt,
dependent_pairFormation_alt,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[as,bs:bag(T)]. \mforall{}[a,b:T].
uiff(bag-remove1(eq;as;a) = bag-remove1(eq;bs;b);((\{a\} + bs) = (\{b\} + as))
\mdownarrow{}\mvee{} ((\mneg{}a \mdownarrow{}\mmember{} as) \mwedge{} (\mneg{}b \mdownarrow{}\mmember{} bs)))
Date html generated:
2019_10_16-AM-11_31_14
Last ObjectModification:
2018_10_11-AM-10_46_07
Theory : bags_2
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