Nuprl Lemma : bag-diff_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[bs,as:bag(T)]. (bag-diff(eq;bs;as) ∈ bag(T)?)
Proof
Definitions occuring in Statement :
bag-diff: bag-diff(eq;bs;as),
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 :
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
top: Top,
all: ∀x:A. B[x],
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
bag-diff: bag-diff(eq;bs;as),
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
sq_type: SQType(T),
guard: {T},
true: True,
false: False,
not: ¬A,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
sq_or: a ↓∨ b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
istype-universe,
deq_wf,
istype-top,
subtype_rel_union,
istype-void,
top_wf,
decide_wf,
bag-remove1_wf,
unit_wf2,
bag_wf,
bag-accum_wf,
bag-remove1-property,
subtype_base_sq,
int_subtype_base,
equal-unit,
bag-append_wf,
single-bag_wf,
bag-member_wf,
bag-remove1-equal,
not_wf,
equal_wf,
squash_wf,
true_wf,
bag-remove1-non-member,
bag-member-append,
subtype_rel_self,
iff_weakening_equal,
equal-wf-T-base,
unit_subtype_base,
it_wf,
equal-wf-base-T
Rules used in proof :
universeEquality,
instantiate,
isectIsTypeImplies,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
applyEquality,
inhabitedIsType,
voidElimination,
isect_memberEquality_alt,
lambdaFormation_alt,
independent_isectElimination,
unionIsType,
inrEquality_alt,
unionElimination,
lambdaEquality_alt,
universeIsType,
inlEquality_alt,
hypothesis,
because_Cache,
unionEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
equalityIsType1,
dependent_functionElimination,
productElimination,
applyLambdaEquality,
promote_hyp,
natural_numberEquality,
cumulativity,
intEquality,
productIsType,
equalityIstype,
functionIsType,
baseClosed,
sqequalBase,
inlFormation_alt,
dependent_set_memberEquality_alt,
independent_pairFormation,
setElimination,
rename,
imageMemberEquality,
hyp_replacement,
imageElimination,
inrFormation_alt,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[bs,as:bag(T)]. (bag-diff(eq;bs;as) \mmember{} bag(T)?)
Date html generated:
2020_05_20-AM-09_04_18
Last ObjectModification:
2020_01_25-AM-10_05_01
Theory : bags_2
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